OEIS/Generating Functions and Recurrences - Revision history

Gfis: combstruct reference

2023-10-20T12:07:09Z

combstruct reference

← Older revision		Revision as of 12:07, 20 October 2023
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	===Cf. [[OEIS/Square Root Recurrences]]===		===Cf. [[OEIS/Square Root Recurrences]]===
			===Further reading===
			* [https://algo.inria.fr/flajolet/Publications/book.pdf Analytic combinatorics] by Philippe Flajolet and Robert Sedgewick. Cambridge: Cambridge University Press, 2009, pp. xiv+810.

2023-10-17T10:28:45Z

Triangles, Riordan arrays, g.f. with two variables: links

← Older revision		Revision as of 10:28, 17 October 2023
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	* [https://oeis.org/A129818 A129818] Riordan array (1/(1+x), x/(1+x)^2)		* [https://oeis.org/A129818 A129818] Riordan array (1/(1+x), x/(1+x)^2)
	O.g.f.: (1+x)/(1+(2-y)*x+x^2).		O.g.f.: (1+x)/(1+(2-y)*x+x^2).
	* '''more''': [~~https://~~https://oeis.org/A260780 A260780] with some diagonals		* [https://oeis.org/A109956 A109956] Inverse of Riordan array (1/(1-x), x/(1-x)^3), [https://oeis.org/A109955 A109955].
			* MMA for exponential Riordan arrays in [https://oeis.org/A256893 A256893]
			* '''more''': [https://oeis.org/A260780 A260780] with some diagonals
	* [https://oeis.org/A207543 A207543] Triangle read by rows, expansion of (1+yx)/(1-2yx+y(y-1)*x^2).		* [https://oeis.org/A207543 A207543] Triangle read by rows, expansion of (1+yx)/(1-2yx+y(y-1)*x^2).
	s := (1+yx)/(1-2yx+y(y-1)*x^2): t := series(s, x, 12):		s := (1+yx)/(1-2yx+y(y-1)*x^2): t := series(s, x, 12):
	seq(print(seq(coeff(coeff(t, x, n), y, m), m=0..n)), n=0..11);		seq(print(seq(coeff(coeff(t, x, n), y, m), m=0..n)), n=0..11);

	===Higher dimensions===		===Higher dimensions===
	* [https://oeis.org/A046816 A046816] Pascal's tetrahedron: entries in 3-dimensional version of Pascal triangle, or trinomial coefficients.		* [https://oeis.org/A046816 A046816] Pascal's tetrahedron: entries in 3-dimensional version of Pascal triangle, or trinomial coefficients.

2023-10-15T17:38:09Z

g.f. for triangle with Maple

← Older revision		Revision as of 17:38, 15 October 2023
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	O.g.f.: (1+x)/(1+(2-y)*x+x^2).		O.g.f.: (1+x)/(1+(2-y)*x+x^2).
	* '''more''': [https://https://oeis.org/A260780 A260780] with some diagonals		* '''more''': [https://https://oeis.org/A260780 A260780] with some diagonals
			* [https://oeis.org/A207543 A207543] Triangle read by rows, expansion of (1+yx)/(1-2yx+y(y-1)*x^2).
			s := (1+yx)/(1-2yx+y(y-1)*x^2): t := series(s, x, 12):
			seq(print(seq(coeff(coeff(t, x, n), y, m), m=0..n)), n=0..11);
	===Higher dimensions===		===Higher dimensions===
	* [https://oeis.org/A046816 A046816] Pascal's tetrahedron: entries in 3-dimensional version of Pascal triangle, or trinomial coefficients.		* [https://oeis.org/A046816 A046816] Pascal's tetrahedron: entries in 3-dimensional version of Pascal triangle, or trinomial coefficients.

2022-11-09T20:32:23Z

Generating functions in your CAS

← Older revision		Revision as of 20:32, 9 November 2022
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	seq(coeff(S, x, n)*n!, n=0..16); # A318599, exponential g.f.		seq(coeff(S, x, n)*n!, n=0..16); # A318599, exponential g.f.

			# A128096
			g:=4(1-z^2-sqrt((1+z+z^2)(1-3z+z^2)))/(1-z+z^2+sqrt((1+z+z^2)(1-3*z+z^2)))^2:
			gser:=series(g, z=0, 38): seq(coeff(gser, z, n), n=1..35);

	read "C:\\Program Files\\Maple 2019\\FPS.mpl":		read "C:\\Program Files\\Maple 2019\\FPS.mpl":
	interface(prettyprint=0):		interface(prettyprint=0):

2021-12-03T09:13:46Z

e.g.f. in Maple

← Older revision		Revision as of 09:13, 3 December 2021
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	ogf := (x^4 + 2x^3 + 7x^2 - 6*x + 1)^(-1/2):		ogf := (x^4 + 2x^3 + 7x^2 - 6*x + 1)^(-1/2):
	series(ogf, x, 27): seq(coeff(%, x, n), n=0..25); # A299443		series(ogf, x, 27): seq(coeff(%, x, n), n=0..25); # A299443

			S:= series((1 + sqrt(1-2sin(2x)))/(2*cos(x)), x, 51):
			seq(coeff(S, x, n)*n!, n=0..16); # A318599, exponential g.f.

	read "C:\\Program Files\\Maple 2019\\FPS.mpl":		read "C:\\Program Files\\Maple 2019\\FPS.mpl":

2021-06-30T08:53:53Z

Generating functions in your CAS: Maple series

← Older revision		Revision as of 08:53, 30 June 2021
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	a := n -> 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], 1):		a := n -> 3^n*hypergeom([-n/3, (1-n)/3, (2-n)/3], [1, 1], 1):
	seq(simplify(a(n)), n=0..24);		seq(simplify(a(n)), n=0..24);

			[https://oreis.org/A007750 A007750] m:=30; s:=series(x(1+6x+x^2)/((1-x)(1-16x^2+x^4)), x, m+1): seq(coeff(s, x, j), j=0..m);
			[https://oreis.org/A007788 A007788] m:=20; s:=series( (1-sin(3x))^(-1/3), x, m+1): seq(j!coeff(s, x, j), j=0..m); # G. C. Greubel
	* Sage		* Sage
	[https://oeis.org/A162760 A162760] (Sage) ((1+x)(1-x^3)/(1-10x+54x^3-45x^4)).series(x, 20).coefficients(x, sparse=False)		[https://oeis.org/A162760 A162760] (Sage) ((1+x)(1-x^3)/(1-10x+54x^3-45x^4)).series(x, 20).coefficients(x, sparse=False)

2021-06-01T19:46:36Z

Generating functions in your CAS: A299443

← Older revision		Revision as of 19:46, 1 June 2021
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	DeleteSpace(convert([seq(coeff(expand((6n-5)(6n-4)(6n-3)(6n-2)(6n-1)n),n,j),j=0..6)],string));		DeleteSpace(convert([seq(coeff(expand((6n-5)(6n-4)(6n-3)(6n-2)(6n-1)n),n,j),j=0..6)],string));

			ogf := (x^4 + 2x^3 + 7x^2 - 6*x + 1)^(-1/2):
			series(ogf, x, 27): seq(coeff(%, x, n), n=0..25); # A299443

	read "C:\\Program Files\\Maple 2019\\FPS.mpl":		read "C:\\Program Files\\Maple 2019\\FPS.mpl":
	interface(prettyprint=0):		interface(prettyprint=0):

2021-05-06T18:56:57Z

Generating functions in your CAS

← Older revision		Revision as of 18:56, 6 May 2021
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	seq(A(n,9), n=0..47); gfun[guessgf]([%],x)[1];		seq(A(n,9), n=0..47); gfun[guessgf]([%],x)[1];
	seq(coeff(series( (1+xexp(x))/(1-x), x, n+1)n!, x, n), n = 0..30);		seq(coeff(series( (1+xexp(x))/(1-x), x, n+1)n!, x, n), n = 0..30);
			with(StringTools):
			DeleteSpace(convert([seq(coeff(expand((6n-5)(6n-4)(6n-3)(6n-2)(6n-1)n),n,j),j=0..6)],string));

	read "C:\\Program Files\\Maple 2019\\FPS.mpl":		read "C:\\Program Files\\Maple 2019\\FPS.mpl":

2021-05-05T08:57:55Z

Holonomic Recurrences

← Older revision		Revision as of 08:57, 5 May 2021
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	(-2+k)a(k)+ka(k+1)+(-1354363k-2708726)a(k+2)		(-2+k)a(k)+ka(k+1)+(-1354363k-2708726)a(k+2)
	* Mathematica ([https://oeis.org/A005325 A005325])		* Mathematica ([https://oeis.org/A005325 A005325])
	RecurrenceTable[{3(-1+n)na[-2+n]+n(1+2n)a[-1+n]-(-5+n)(7+n)a[n]==0, a[5]==1,a[6]==6}, a,{n,5,20}		RecurrenceTable[{3(-1+n)na[-2+n]+n(1+2n)a[-1+n]-(-5+n)(7+n)a[n]==0, a[5]==1,a[6]==6}, a,{n,5,20}]

	===Cf. [[OEIS/Square Root Recurrences]]===		===Cf. [[OEIS/Square Root Recurrences]]===

2021-05-01T13:21:08Z

Generating functions in your CAS: 3rd PARI

← Older revision		Revision as of 13:21, 1 May 2021
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	[https://oeis.org/A162760 A162760] (PARI) my(x='x+O('x^20)); Vec((1+x)(1-x^3)/(1-10x+54x^3-45x^4)) \\ _G. C. Greubel_, Apr 26 2019		[https://oeis.org/A162760 A162760] (PARI) my(x='x+O('x^20)); Vec((1+x)(1-x^3)/(1-10x+54x^3-45x^4)) \\ _G. C. Greubel_, Apr 26 2019
	[https://oeis.org/A025225 A025225]: a(n)=polcoeff((1-sqrt(1-8x+xO(x^n)))/2, n)		[https://oeis.org/A025225 A025225]: a(n)=polcoeff((1-sqrt(1-8x+xO(x^n)))/2, n)
			for(n=0,32,print(n" ",polcoeff(x(1-2x^2*(1+x-x^3-x^4-x^5)+O(x^n))/(1+x),n)))
	* Mathematica		* Mathematica
	CoefficientList[Series[polx1/polx2, {x, 0, 29}], x]		CoefficientList[Series[polx1/polx2, {x, 0, 29}], x]