Expansion of polynomials - Revision history

Gfis: pow, seqs

2025-06-11T08:23:22Z

pow, seqs

← Older revision		Revision as of 08:23, 11 June 2025
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	====Special operators====		====Special operators====
	* <code>A</code> push the generating function to be defined		* <code>A</code> push the generating function to be defined
	* <code>S, T, U, V</code> push the 1st, 2nd, 3rd, 4th generating function from the list of additional sequences		* <code>B, C, D, E</code> push the 1st, 2nd, 3rd, 4th generating function from the list of additional source sequences
	* <code>x</code> push the free variable		* <code>x</code> push the free variable
			* <code>q</code> same as x, for eta products
	* <code>0, 1, 2, ... </code> push an integer		* <code>0, 1, 2, ... </code> push an integer
	* <code>n</code> push the current index		* <code>n</code> push the current index
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	Replace the 2 top stack elements '''a''', '''b''' by '''a op b''', where ''op'' is one of:		Replace the 2 top stack elements '''a''', '''b''' by '''a op b''', where ''op'' is one of:
	* <code>+</code> <code>-</code> <code>*</code> <code>/</code>		* <code>+</code> <code>-</code> <code>*</code> <code>/</code>
			* <code>pow</code> - in contrast to <code>^i</code>, the exponent may be a polynomial; <code>a pow b</code> is implemented by <code>exp(log(a)*b)</code>
	* <code>agm</code>		* <code>agm</code>
	* <code>besselI</code>		* <code>besselI -</code> only <code>besselI(0,x)</code> is currently working
	====Functions with 1 argument====		====Functions with 1 argument====
	Replace the top stack element by the function's result:		Replace the top stack element by the function's result:
	* <code>abs</code> <code>acosh</code> <code>asin</code> <code>asinh</code> <code>atan</code> <code>atanh</code> <code>cos</code> <code>cosh</code> <code>cot</code> <code>coth</code> <code>csc</code> <code>csch</code> <code>ellipticD</code> <code>ellipticE</code> <code>ellipticK eta</code> <code>exp</code> <code>lambertW</code> <code>log</code> <code>neg</code> <code>sec</code> <code>sech</code> <code>sin</code> <code>sinh</code> <code>sqrt</code> <code>tan</code> <code>tanh</code>		* <code>abs</code> <code>acosh</code> <code>asin</code> <code>asinh</code> <code>atan</code> <code>atanh</code> <code>cos</code> <code>cosh</code> <code>cot</code> <code>coth</code> <code>csc</code> <code>csch</code> <code>ellipticD</code> <code>ellipticE</code> <code>ellipticK eta</code> <code>exp</code> <code>lambertW</code> <code>log</code> <code>neg</code> <code>sec</code> <code>sech</code> <code>sin</code> <code>sinh</code> <code>sqrt</code> <code>tan</code> <code>tanh</code>
	* <code>catalan</code>(x) = (1 - sqrt(1 - 4x)) / (2x)		* <code>catalan</code>(x) = (1 - sqrt(1 - 4x)) / (2x)

	==== ~~Subordinate~~ generating functions ====		==== Additional source sequences as generating functions ====
	Any sequence implemented in jOEIS can be used as a subordinate generating function. Usual examples are the [[OEIS/Eta products\|'''Eta Products''']] ~~and~~ '''theta_3''' = A000122, '''theta_4''' = A002448.		Any sequence implemented in jOEIS can be used as a subordinate generating function. Usual examples are the

			* [[OEIS/Eta products\|'''Eta Products''']]
			* '''theta_3''' = [[oeis:A000122\|A000122]]
			* '''theta_4''' = [[oeis:A002448\|A002448]]
			* '''catalan''' = [[oeis:A000108\|A000108]] <code>= (1 - sqrt(1 - 4x)) / (2x)</code>

	===Caveats===		===Caveats===

Gfis: Eta products

2025-05-30T21:54:47Z

Eta products

← Older revision		Revision as of 21:54, 30 May 2025
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	**
	===Parameters===		===Parameters===
	The expansion of (exponential) generating functions is performed by <code>'''[https://github.com/archmageirvine/joeis/tree/master/src/irvine/oeis/PolynomialFieldSequence.java PolynomialFieldSequence]'''</code> with ~~a constructor~~ that ~~has~~ the following parameters:		The expansion of (exponential) generating functions is performed by <code>'''[https://github.com/archmageirvine/joeis/tree/master/src/irvine/oeis/PolynomialFieldSequence.java PolynomialFieldSequence]'''</code> with constructors that have the following parameters:
	* <code>offset:</code> first index		* <code>offset:</code> first index
	* <code>polys:</code> a list of list of integers in square brackets, specifying the coefficients of a list of polynomials.		* <code>polys:</code> a list of list of integers in square brackets, specifying the coefficients of a list of polynomials.
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	* <code>dist:</code> an optional increment to the current index		* <code>dist:</code> an optional increment to the current index
	* <code>gfType:</code> 0 for an o.g.f., 1 for an e.g.f., +4 if the denominator should be returned instead of the numerator (default 0 = o.g.f)		* <code>gfType:</code> 0 for an o.g.f., 1 for an e.g.f., +4 if the denominator should be returned instead of the numerator (default 0 = o.g.f)
	* <code>modulus, ~~factor~~:</code> which element to take when zeros are suppressed ~~(default 1)~~, ~~and periodicity when zeroes are suppressed (default 1). Even~~ coefficients are selected by (0,2), odd coefficients by (1,2).		* <code>modulus, periodicity:</code> which element to take when zeros are to be suppressed. For example, even coefficients are selected by (0,2) and odd coefficients by (1,2).
	* <code>seqs:</code> a variable number of sequences defining ~~other~~ generating functions		* <code>seqs:</code> a variable number of sequences defining subordinate generating functions

	==== Generator Callcodes ====		==== Generator Callcodes ====
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	* <code>abs</code> <code>acosh</code> <code>asin</code> <code>asinh</code> <code>atan</code> <code>atanh</code> <code>cos</code> <code>cosh</code> <code>cot</code> <code>coth</code> <code>csc</code> <code>csch</code> <code>ellipticD</code> <code>ellipticE</code> <code>ellipticK eta</code> <code>exp</code> <code>lambertW</code> <code>log</code> <code>neg</code> <code>sec</code> <code>sech</code> <code>sin</code> <code>sinh</code> <code>sqrt</code> <code>tan</code> <code>tanh</code>		* <code>abs</code> <code>acosh</code> <code>asin</code> <code>asinh</code> <code>atan</code> <code>atanh</code> <code>cos</code> <code>cosh</code> <code>cot</code> <code>coth</code> <code>csc</code> <code>csch</code> <code>ellipticD</code> <code>ellipticE</code> <code>ellipticK eta</code> <code>exp</code> <code>lambertW</code> <code>log</code> <code>neg</code> <code>sec</code> <code>sech</code> <code>sin</code> <code>sinh</code> <code>sqrt</code> <code>tan</code> <code>tanh</code>
	* <code>catalan</code>(x) = (1 - sqrt(1 - 4x)) / (2x)		* <code>catalan</code>(x) = (1 - sqrt(1 - 4x)) / (2x)

			==== Subordinate generating functions ====
			Any sequence implemented in jOEIS can be used as a subordinate generating function. Usual examples are the [[OEIS/Eta products\|'''Eta Products''']] and '''theta_3''' = A000122, '''theta_4''' = A002448.

	===Caveats===		===Caveats===
	* Usually the g.f.s have offset 0 and start with a term 1, so <code>p0 = "[[1]]"</code> in most cases. The first polynomial p0 must always include the coefficient of x<sup>0</sup>, even if the offset is > 0, for example: <code>p0 = "[[0,1]]"</code>.		* Usually the g.f.s have offset 0 and start with a term 1, so <code>p0 = "[[1]]"</code> in most cases. The first polynomial p0 must always include the coefficient of x<sup>0</sup>, even if the offset is > 0, for example: <code>p0 = "[[0,1]]"</code>.

Gfis: /* Examples: */

2025-05-30T21:43:06Z

Examples:

← Older revision		Revision as of 21:43, 30 May 2025
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	==== Examples: ====		==== Examples: ====
			* '''[[oeis:A308847\|A308847]]''' Expansion of e.g.f. exp(-2x) / BesselI(0,2x).
	* A308847 Expansion of e.g.f. exp(-2x) / BesselI(0,2x).		A-number callcode offset p0=1,p1=2x,p2=-2x exp(p2)/besselI(0,p1) dist gftype=exp
			A308847 polyx 0 "[1],[0,2],[0,-2]" "p2,exp,0,p1,besselI,/" 0 1
	~~<code>~~A-number callcode offset p0=1,p1=2x,p2=-2x exp(p2)/besselI(0,p1) dist gftype=exp~~</code>~~
	~~<code>~~A308847 polyx 0 "[1],[0,2],[0,-2]" "p2,exp,0,p1,besselI,/" 0 1~~</code>~~

	'''[[oeis:A295237\|A295237]]''' <code>a:=series(csc(x)(1-sqrt(1-4sin(x)))/2, x=0, 20): seq(n!coeff(a, x, n), n=0..19); # E.g.f.</code>
	* '''[[oeis:A345969\|A345969]]''' Expansion of the e.g.f. <code>1 / sqrt(3 - 2 / ((1 - x)*exp(x)))</code>		* '''[[oeis:A345969\|A345969]]''' Expansion of the e.g.f. <code>1 / sqrt(3 - 2 / ((1 - x)*exp(x)))</code>
	** Maple: <code>a := series(1/sqrt(3-2/((1-x)exp(x))), x=0, 25): seq(n!coeff(a, x, n), n=0..24);</code>		** Maple: <code>a := series(1/sqrt(3-2/((1-x)exp(x))), x=0, 25): seq(n!coeff(a, x, n), n=0..24);</code>
	** jOEIS: <code>new PolynomialFieldSequence(0, "[[1],[1,-1]]" ",1,3,2,p1,x,exp,*,/,-,sqrt,/", 0, 1)</code>		** jOEIS: <code>new PolynomialFieldSequence(0, "[[1],[1,-1]]" ",1,3,2,p1,x,exp,*,/,-,sqrt,/", 0, 1)</code>

	=== Postfix notation ===		=== Postfix notation ===

Gfis: Reordered

2025-05-30T21:40:53Z

Reordered

← Older revision		Revision as of 21:40, 30 May 2025
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	* ~~'''[[oeis:A295237\|A295237]]''' <code>a:=series(csc(x)~~~~(1-sqrt(1-4sin(x)))/2, x=0, 20): seq(n!*coeff(a, x, n), n=0..19); # E.g.f.</code>~~		**
	* '''[[oeis:A345969\|A345969]]''' Expansion of the e.g.f. <code>1 / sqrt(3 - 2 / ((1 - x)*exp(x)))</code>
	** Maple: <code>a := series(1/sqrt(3-2/((1-x)exp(x))), x=0, 25): seq(n!coeff(a, x, n), n=0..24);</code>
	** jOEIS: <code>new PolynomialFieldSequence(0, "[[1],[1,-1]]" ",1,3,2,p1,x,exp,*,/,-,sqrt,/", 0, 1)</code>
	===Parameters===		===Parameters===
	The expansion of (exponential) generating functions is performed by <code>'''[https://github.com/archmageirvine/joeis/tree/master/src/irvine/oeis/PolynomialFieldSequence.java PolynomialFieldSequence]'''</code> with a constructor that has ~~up to 5~~ parameters:		The expansion of (exponential) generating functions is performed by <code>'''[https://github.com/archmageirvine/joeis/tree/master/src/irvine/oeis/PolynomialFieldSequence.java PolynomialFieldSequence]'''</code> with a constructor that has the following parameters:
	* <code>offset:</code> first index		* <code>offset:</code> first index
	* <code>polys:</code> a list of list of integers in square brackets, specifying the coefficients of a list of polynomials~~. The polynomials are referenced by p0, p1, p2, ... where p0 gives the initial term of the iteration. For example, 1-x = '''1'''x^0 '''-1'''x^1 -> ['''1,-1''']~~.		* <code>polys:</code> a list of list of integers in square brackets, specifying the coefficients of a list of polynomials.
	* <code>postfix:</code> a string starting with a delimiter, followed by a list of elements that are separated by this delimiter		* <code>postfix:</code> a string starting with a delimiter, followed by a list of elements that are separated by this delimiter
	* <code>dist:</code> an optional increment to the current index		* <code>dist:</code> an optional increment to the current index
	* <code>gfType:</code> 0 for an o.g.f., 1 for an e.g.f., +4 if the denominator should be returned instead of the numerator (default 0 = o.g.f)		* <code>gfType:</code> 0 for an o.g.f., 1 for an e.g.f., +4 if the denominator should be returned instead of the numerator (default 0 = o.g.f)
	* <code>modulus, factor:</code> which element to take when zeros are suppressed (default 1), and periodicity when zeroes are suppressed (default 1). Even coefficients are selected by (0,2), odd coefficients by (1,2).		* <code>modulus, factor:</code> which element to take when zeros are suppressed (default 1), and periodicity when zeroes are suppressed (default 1). Even coefficients are selected by (0,2), odd coefficients by (1,2).
			* <code>seqs:</code> a variable number of sequences defining other generating functions

			==== Generator Callcodes ====
			The following callcodes of the generator use the specified parameter combinations:
			poly (offset, polys, postfix)
			polyx (offset, polys, postfix, dist, gftype)
			polyz (offset, polys, postfix, dist, gftype, modulus, factor)
			polya (offset, polys, postfix, seqs)
			polyxa (offset, polys, postfix, dist, gftype, seqs)
			polyza (offset, polys, postfix, dist, gftype, modulus, factor, seqs)

			==== Polynomials ====
			The polynomials in polys are referenced by p0, p1, p2, ... where p0 gives the initial term of the iteration. A single polynomial is specified by the list of coefficients with increasing exponents, for example, 1-x = '''1'''x^0 '''-1'''x^1 -> ['''1,-1''']. Most often the initial <code>p0 = "[1]" = 1.</code>
			The list of polynomials may optionally (for testing) be followed by a list of sequence numbers for subordinate generating functions, but normally such sequences should be specified as a separate trailign list of sequence instances.

			==== Examples: ====

			* A308847 Expansion of e.g.f. exp(-2x) / BesselI(0,2x).

			<code>A-number callcode offset p0=1,p1=2x,p2=-2x exp(p2)/besselI(0,p1) dist gftype=exp</code>
			<code>A308847 polyx 0 "[1],[0,2],[0,-2]" "p2,exp,0,p1,besselI,/" 0 1</code>

			'''[[oeis:A295237\|A295237]]''' <code>a:=series(csc(x)(1-sqrt(1-4sin(x)))/2, x=0, 20): seq(n!coeff(a, x, n), n=0..19); # E.g.f.</code>
			* '''[[oeis:A345969\|A345969]]''' Expansion of the e.g.f. <code>1 / sqrt(3 - 2 / ((1 - x)*exp(x)))</code>
			** Maple: <code>a := series(1/sqrt(3-2/((1-x)exp(x))), x=0, 25): seq(n!coeff(a, x, n), n=0..24);</code>
			** jOEIS: <code>new PolynomialFieldSequence(0, "[[1],[1,-1]]" ",1,3,2,p1,x,exp,*,/,-,sqrt,/", 0, 1)</code>

	=== Postfix notation ===		=== Postfix notation ===
	The operations of the postfix string are performed in their sequential order. They operate on a stack, and the final result is the lowest stack element.		The operations of the postfix string are performed in their sequential order. They operate on a stack, and the final result is the lowest stack element.
	====Special operators====		====Special operators====
	* <code>A</code> push the generating function to be defined		* <code>A</code> push the generating function to be defined
			* <code>S, T, U, V</code> push the 1st, 2nd, 3rd, 4th generating function from the list of additional sequences
	* <code>x</code> push the free variable		* <code>x</code> push the free variable
	* <code>0, 1, 2, ... </code> push an integer		* <code>0, 1, 2, ... </code> push an integer
			* <code>n</code> push the current index
	* <code>p<sub>i</sub></code> push the predefined polynomial with index '''i''' (cf. <code>polys</code> above)		* <code>p<sub>i</sub></code> push the predefined polynomial with index '''i''' (cf. <code>polys</code> above)
	* <code><i</code> shift the top stack element by '''i''', i.e. multiply by x<sup>i</sup>		* <code><i</code> shift the top stack element by '''i''', i.e. multiply by x<sup>i</sup>
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	* <code>+</code> <code>-</code> <code>*</code> <code>/</code>		* <code>+</code> <code>-</code> <code>*</code> <code>/</code>
	* <code>agm</code>		* <code>agm</code>
			* <code>besselI</code>
	====Functions with 1 argument====		====Functions with 1 argument====
	Replace the top stack element by the function's result:		Replace the top stack element by the function's result:
	* <code>abs</code> <code>acosh</code> <code>asin</code> <code>asinh</code> <code>atan</code> <code>atanh</code> <code>cos</code> <code>cosh</code> <code>cot</code> <code>coth</code> <code>csc</code> <code>csch</code> <code>eta</code> <code>exp</code> <code>lambertW</code> <code>log</code> <code>neg</code> <code>sec</code> <code>sech</code> <code>sin</code> <code>sinh</code> <code>sqrt</code> <code>tan</code> <code>tanh</code>		* <code>abs</code> <code>acosh</code> <code>asin</code> <code>asinh</code> <code>atan</code> <code>atanh</code> <code>cos</code> <code>cosh</code> <code>cot</code> <code>coth</code> <code>csc</code> <code>csch</code> <code>ellipticD</code> <code>ellipticE</code> <code>ellipticK eta</code> <code>exp</code> <code>lambertW</code> <code>log</code> <code>neg</code> <code>sec</code> <code>sech</code> <code>sin</code> <code>sinh</code> <code>sqrt</code> <code>tan</code> <code>tanh</code>
	* <code>catalan</code>(x) = (1 - sqrt(1 - 4x)) / (2x)		* <code>catalan</code>(x) = (1 - sqrt(1 - 4x)) / (2x)
	===Caveats===		===Caveats===
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	* <code>sqrt</code> expects an argument of the form ''1-x''. sqrt(x) does not work, it can sometimes be avoided by quadrating all occurrences of x.		* <code>sqrt</code> expects an argument of the form ''1-x''. sqrt(x) does not work, it can sometimes be avoided by quadrating all occurrences of x.
	===Commandline activation===		===Commandline activation===
	A script <code>OEIS-mat/scripts/poly</code> facilitates the testing of the polynomial expansions. It is activated with ~~up to 4~~ parameters:		A script <code>OEIS-mat/scripts/poly</code> facilitates the testing of the polynomial expansions. It is activated with the following parameters and options:
	* <code>poly polys postfix [-i dist] [-t gfTtype]~~</code>~~
	~~===Generator Callcodes===~~		<code>poly polys postfix [-i dist] [-t gfTtype] [-n number of terms] [-o offset] [-d debug mode]</code>
	~~There are 3 callcodes:~~
	* <code>poly offset ~~polys postfix</code>~~
	* <code>polyx offset polys postfix dist gfType</code>
	* <code>polyz offset polys postfix dist gfType modulus factor</code>

Gfis: /* Functions with 1 argument */

2025-04-25T19:57:41Z

Functions with 1 argument

← Older revision		Revision as of 19:57, 25 April 2025
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	* <code>agm</code>		* <code>agm</code>
	====Functions with 1 argument====		====Functions with 1 argument====
	Replace the top stack element		Replace the top stack element by the function's result:
	* <code>abs</code> <code>acosh</code> <code>asin</code> <code>asinh</code> <code>atan</code> <code>atanh</code> <code>cos</code> <code>cosh</code> <code>cot</code> <code>coth</code> <code>csc</code> <code>csch</code> <code>eta</code> <code>exp</code> <code>lambertW</code> <code>log</code> <code>neg</code> <code>sec</code> <code>sech</code> <code>sin</code> <code>sinh</code> <code>sqrt</code> <code>tan</code> <code>tanh</code>		* <code>abs</code> <code>acosh</code> <code>asin</code> <code>asinh</code> <code>atan</code> <code>atanh</code> <code>cos</code> <code>cosh</code> <code>cot</code> <code>coth</code> <code>csc</code> <code>csch</code> <code>eta</code> <code>exp</code> <code>lambertW</code> <code>log</code> <code>neg</code> <code>sec</code> <code>sech</code> <code>sin</code> <code>sinh</code> <code>sqrt</code> <code>tan</code> <code>tanh</code>
	* <code>catalan</code>(x) = (1 - sqrt(1 - 4x)) / (2x)		* <code>catalan</code>(x) = (1 - sqrt(1 - 4x)) / (2x)

Gfis at 19:56, 25 April 2025

2025-04-25T19:56:52Z

← Older revision		Revision as of 19:56, 25 April 2025
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	* '''[[oeis:A295237\|A295237]]''' <code>a:=series(csc(x)(1-sqrt(1-4sin(x)))/2, x=0, 20): seq(n!*coeff(a, x, n), n=0..19); # E.g.f.</code>		* '''[[oeis:A295237\|A295237]]''' <code>a:=series(csc(x)(1-sqrt(1-4sin(x)))/2, x=0, 20): seq(n!*coeff(a, x, n), n=0..19); # E.g.f.</code>
	* '''[[oeis:A345969\|A345969]]''' Expansion of the e.g.f. 1 / sqrt(3 - 2 / ((1 - x)exp(x))). Maple: <code>a := series(1/sqrt(3-2/((1-x)exp(x))), x=0, 25): seq(n!coeff(a, x, n), n=0..24);</code> jOEIS: <code>new PolynomialFieldSequence(0, "[[1],[1,-1]]" ",1,3,2,p1,x,exp,,/,-,sqrt,/", 0, 1)</code>		* '''[[oeis:A345969\|A345969]]''' Expansion of the e.g.f. <code>1 / sqrt(3 - 2 / ((1 - x)*exp(x)))</code>
			** Maple: <code>a := series(1/sqrt(3-2/((1-x)exp(x))), x=0, 25): seq(n!coeff(a, x, n), n=0..24);</code>
			** jOEIS: <code>new PolynomialFieldSequence(0, "[[1],[1,-1]]" ",1,3,2,p1,x,exp,*,/,-,sqrt,/", 0, 1)</code>
	===Parameters===		===Parameters===
	The expansion of (exponential) generating functions is performed by <code>'''[https://github.com/archmageirvine/joeis/tree/master/src/irvine/oeis/PolynomialFieldSequence.java PolynomialFieldSequence]'''</code> with a constructor that has up to 5 parameters:		The expansion of (exponential) generating functions is performed by <code>'''[https://github.com/archmageirvine/joeis/tree/master/src/irvine/oeis/PolynomialFieldSequence.java PolynomialFieldSequence]'''</code> with a constructor that has up to 5 parameters:

Gfis: /* Caveats */

2025-04-25T19:54:43Z

Caveats

← Older revision		Revision as of 19:54, 25 April 2025
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	* Usually the g.f.s have offset 0 and start with a term 1, so <code>p0 = "[[1]]"</code> in most cases. The first polynomial p0 must always include the coefficient of x<sup>0</sup>, even if the offset is > 0, for example: <code>p0 = "[[0,1]]"</code>.		* Usually the g.f.s have offset 0 and start with a term 1, so <code>p0 = "[[1]]"</code> in most cases. The first polynomial p0 must always include the coefficient of x<sup>0</sup>, even if the offset is > 0, for example: <code>p0 = "[[0,1]]"</code>.
	* Usually the parameter <code>dist</code> is not needed. Sometimes, it must be set to a value > 0 to start the iteration properly.		* Usually the parameter <code>dist</code> is not needed. Sometimes, it must be set to a value > 0 to start the iteration properly.
	* <code>sqrt</code> expects an argument of the form ''1-x''. sqrt(x) does not work, it can ~~sometime~~ be avoided by quadrating all occurrences of x.		* <code>sqrt</code> expects an argument of the form ''1-x''. sqrt(x) does not work, it can sometimes be avoided by quadrating all occurrences of x.
	===Commandline activation===		===Commandline activation===
	A script <code>OEIS-mat/scripts/poly</code> facilitates the testing of the polynomial expansions. It is activated with up to 4 parameters:		A script <code>OEIS-mat/scripts/poly</code> facilitates the testing of the polynomial expansions. It is activated with up to 4 parameters:

Gfis: Caveats, commandline, callcodes

2025-04-25T19:54:09Z

Caveats, commandline, callcodes

@@ Line 17: / Line 17: @@
 * <code>p<sub>i</sub></code> push the predefined polynomial with index '''i''' (cf. <code>polys</code> above)
 * <code>&lt;i</code> shift the top stack element by '''i''', i.e. multiply by x<sup>i</sup>
-* <code>^i</code> take the top stack element to the i-th power
+* <code>^i</code> take the top stack element to the i-th power, where <code>i</code> can be an integer or a fraction (<code>^1/2</code> is <code>sqrt</code>)
 * <code>dif</code> differentiate the top stack element
 * <code>int</code> formal integration of the top stack element
@@ Line 29: / Line 29: @@
 * <code>abs</code> <code>acosh</code> <code>asin</code> <code>asinh</code> <code>atan</code> <code>atanh</code>  <code>cos</code> <code>cosh</code> <code>cot</code> <code>coth</code> <code>csc</code> <code>csch</code> <code>eta</code> <code>exp</code> <code>lambertW</code> <code>log</code> <code>neg</code> <code>sec</code> <code>sech</code> <code>sin</code> <code>sinh</code> <code>sqrt</code> <code>tan</code> <code>tanh</code>
 * <code>catalan</code>(x) = (1 - sqrt(1 - 4*x)) / (2*x)

Gfis: postfix operations

2025-04-25T19:34:11Z

postfix operations

@@ Line 1: / Line 1: @@
 * '''[[oeis:A295237|A295237]]''' <code>a:=series(csc(x)*(1-sqrt(1-4*sin(x)))/2, x=0, 20): seq(n!*coeff(a, x, n), n=0..19); # E.g.f.</code>
 * '''[[oeis:A345969|A345969]]''' Expansion of the e.g.f. 1 / sqrt(3 - 2 / ((1 - x)*exp(x))). Maple: <code>a := series(1/sqrt(3-2/((1-x)*exp(x))), x=0, 25): seq(n!*coeff(a, x, n), n=0..24);</code> jOEIS: <code>new PolynomialFieldSequence(0, "[[1],[1,-1]]" ",1,3,2,p1,x,exp,*,/,-,sqrt,/", 0, 1)</code>
+===Parameters===
 The expansion of (exponential) generating functions is performed by <code>'''[https://github.com/archmageirvine/joeis/tree/master/src/irvine/oeis/PolynomialFieldSequence.java PolynomialFieldSequence]'''</code> with a constructor that has up to 5 parameters:
 * <code>offset:</code> first index
 * <code>polys:</code> a list of list of integers in square brackets, specifying the coefficients of a list of polynomials. The polynomials are referenced by p0, p1, p2, ... where p0 gives the initial term of the iteration. For example, 1-x = '''1'''*x^0 '''-1'''*x^1 -> ['''1,-1'''].
@@ Line 11: / Line 9: @@
 * <code>gfType:</code> 0 for an o.g.f., 1 for an e.g.f., +4 if the denominator should be returned instead of the numerator (default 0 = o.g.f)
 * <code>modulus, factor:</code> which element to take when zeros are suppressed (default 1), and periodicity when zeroes are suppressed (default 1). Even coefficients are selected by (0,2), odd coefficients by (1,2).

Gfis: A345969

2025-03-09T13:19:05Z

A345969

← Older revision		Revision as of 13:19, 9 March 2025
Line 1:		Line 1:
	* '''[[oeis:A295237\|A295237]]''' <code>a:=series(csc(x)(1-sqrt(1-4sin(x)))/2, x=0, 20): seq(n!*coeff(a, x, n), n=0..19); # E.g.f.</code>		* '''[[oeis:A295237\|A295237]]''' <code>a:=series(csc(x)(1-sqrt(1-4sin(x)))/2, x=0, 20): seq(n!*coeff(a, x, n), n=0..19); # E.g.f.</code>
			* '''[[oeis:A345969\|A345969]]''' Expansion of the e.g.f. 1 / sqrt(3 - 2 / ((1 - x)exp(x))). Maple: <code>a := series(1/sqrt(3-2/((1-x)exp(x))), x=0, 25): seq(n!coeff(a, x, n), n=0..24);</code> jOEIS: <code>new PolynomialFieldSequence(0, "[[1],[1,-1]]" ",1,3,2,p1,x,exp,,/,-,sqrt,/", 0, 1)</code>

	==== Postfix notation ====		==== Postfix notation ====
	The expansion of (exponential) generating functions is performed by <code>'''[https://github.com/archmageirvine/joeis/tree/master/src/irvine/oeis/PolynomialFieldSequence.java PolynomialFieldSequence]'''</code> with a constructor that has up to 5 parameters:		The expansion of (exponential) generating functions is performed by <code>'''[https://github.com/archmageirvine/joeis/tree/master/src/irvine/oeis/PolynomialFieldSequence.java PolynomialFieldSequence]'''</code> with a constructor that has up to 5 parameters:

	* <code>offset</code>: first index		* <code>offset:</code> first index
	* <code>polys</code>: a list of list of integers in square brackets, specifying the coefficients of a list of polynomials. The polynomials are referenced by p0, p1, p2, ... where p0 gives the initial term of the iteration.		* <code>polys:</code> a list of list of integers in square brackets, specifying the coefficients of a list of polynomials. The polynomials are referenced by p0, p1, p2, ... where p0 gives the initial term of the iteration. For example, 1-x = '''1'''x^0 '''-1'''x^1 -> ['''1,-1'''].
	* <code>postfix</code>: a string starting with a delimiter, followed by a list of elements that are separated by this delimiter		* <code>postfix:</code> a string starting with a delimiter, followed by a list of elements that are separated by this delimiter
	* <code>dist</code>: an optional increment to the current index		* <code>dist:</code> an optional increment to the current index
	* <code>gfType</code>: 0 for an o.g.f., 1 for an e.g.f., +4 if the denominator should be returned instead of the numerator, default 0		* <code>gfType:</code> 0 for an o.g.f., 1 for an e.g.f., +4 if the denominator should be returned instead of the numerator (default 0 = o.g.f)
	* <code>modulus</code>: which element to take when zeros are suppressed, default 1		* <code>modulus, factor:</code> which element to take when zeros are suppressed (default 1), and periodicity when zeroes are suppressed (default 1). Even coefficients are selected by (0,2), odd coefficients by (1,2).
	* <code>factor</code>: periodicity when zeroes are suppressed, default 1. Even coefficients are selected by (0,2), odd coefficients by (1,2).