Expansion of polynomials: Difference between revisions
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* '''[[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: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> | |||
==== 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> | * <code>offset:</code> first index | ||
* <code>polys</code> | * <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> | * <code>postfix:</code> a string starting with a delimiter, followed by a list of elements that are separated by this delimiter | ||
* <code>dist</code> | * <code>dist:</code> an optional increment to the current index | ||
* <code>gfType</code> | * <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> | * <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). | ||
Latest revision as of 13:19, 9 March 2025
- A295237
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. - A345969 Expansion of the e.g.f. 1 / sqrt(3 - 2 / ((1 - x)*exp(x))). Maple:
a := series(1/sqrt(3-2/((1-x)*exp(x))), x=0, 25): seq(n!*coeff(a, x, n), n=0..24);jOEIS:new PolynomialFieldSequence(0, "[[1],[1,-1]]" ",1,3,2,p1,x,exp,*,/,-,sqrt,/", 0, 1)
Postfix notation
The expansion of (exponential) generating functions is performed by PolynomialFieldSequence with a constructor that has up to 5 parameters:
offset:first indexpolys: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].postfix:a string starting with a delimiter, followed by a list of elements that are separated by this delimiterdist:an optional increment to the current indexgfType: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)modulus, factor: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).