cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A361772 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(2*n-1).

Original entry on oeis.org

1, 1, 8, 61, 600, 6072, 65804, 733435, 8415694, 98529785, 1173278329, 14162417506, 172914841649, 2131621288494, 26495818020038, 331706510158239, 4178800564364333, 52935845003315662, 673878770026778330, 8616336680850069832, 110606714769468383785, 1424933340070339610543
Offset: 0

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Author

Paul D. Hanna, May 13 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + 8*x^2 + 61*x^3 + 600*x^4 + 6072*x^5 + 65804*x^6 + 733435*x^7 + 8415694*x^8 + 98529785*x^9 + 1173278329*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
    A[#A] = polcoeff( sum(m=-#A, #A, x^m * (2*Ser(A) - (-x)^m)^(2*m-1) ), #A-1)/2); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(2*n-1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(2*n^2) / (1 - 2*A(x)*(-x)^n)^(2*n+1).

A361773 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(3*n-1).

Original entry on oeis.org

1, 2, 34, 677, 15660, 393790, 10433402, 286990626, 8117763488, 234635708480, 6899771599141, 205768408153474, 6208628685564955, 189188990142419693, 5813805339043713267, 179968235623379467274, 5606627898452185950618, 175650401043239524832783, 5530500462355496324862920
Offset: 0

Views

Author

Paul D. Hanna, May 13 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 34*x^2 + 677*x^3 + 15660*x^4 + 393790*x^5 + 10433402*x^6 + 286990626*x^7 + 8117763488*x^8 + 234635708480*x^9 + 6899771599141*x^10 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
    A[#A] = polcoeff( sum(m=-#A, #A, x^m * (2*Ser(A) - (-x)^m)^(3*m-1) ), #A-1)/2); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(3*n-1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(3*n^2) / (1 - 2*A(x)*(-x)^n)^(3*n+1).

A361771 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(n-1).

Original entry on oeis.org

1, 1, 1, 7, 28, 89, 421, 1898, 7912, 36412, 169960, 779139, 3668210, 17486938, 83333003, 400956919, 1943928504, 9455346485, 46225027071, 227066384875, 1119123274755, 5534782142253, 27463607765186, 136652474592260, 681728348606011, 3409395265172439, 17088672210734316
Offset: 0

Views

Author

Paul D. Hanna, May 13 2023

Keywords

Examples

			G.f.: A(x) = 1 + x + x^2 + 7*x^3 + 28*x^4 + 89*x^5 + 421*x^6 + 1898*x^7 + 7912*x^8 + 36412*x^9 + 169960*x^10 + 779139*x^11 + 3668210*x^12 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
    A[#A] = polcoeff( sum(m=-#A, #A, x^m * (2*Ser(A) - (-x)^m)^(m-1) ), #A-1)/2); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n may be defined by the following.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - (-x)^n)^(n-1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n^2) / (1 - 2*A(x)*(-x)^n)^(n+1).

A363114 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(4*n-1).

Original entry on oeis.org

1, 4, 138, 6571, 353935, 20694945, 1276853497, 81834405039, 5395444806588, 363600236084796, 24933767742193052, 1734273108108910743, 122058422998192278797, 8676376795137864622232, 622018188741046650309066, 44922343315319150402783783, 3265215115112327274815579250
Offset: 0

Views

Author

Paul D. Hanna, May 14 2023

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 138*x^2 + 6571*x^3 + 353935*x^4 + 20694945*x^5 + 1276853497*x^6 + 81834405039*x^7 + 5395444806588*x^8 + ...
		

Crossrefs

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
    A[#A] = polcoeff(-1 + sum(m=-#A, #A, x^m * (2*Ser(A) - x^m)^(4*m-1) ), #A-1)/2); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))
    
  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
    A[#A] = polcoeff(1 - sum(m=-#A, #A, x^(4*m^2)/(1 - 2*Ser(A)*x^m)^(4*m+1) ), #A-1)/2); A[n+1]}
    for(n=0, 30, print1(a(n), ", "))

Formula

Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(4*n-1).
(2) -1 = Sum_{n=-oo..+oo} x^(4*n^2) / (1 - 2*A(x)*x^n)^(4*n+1).
Showing 1-4 of 4 results.