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.

Previous Showing 11-17 of 17 results.

A358959 a(n) = coefficient of x^n in A(x) such that: 0 = Sum_{n=-oo..+oo} x^(9*n) * (x^n - 2*A(x))^(10*n+1).

Original entry on oeis.org

1, 9, 171, 3819, 94221, 2474541, 67842255, 1919233719, 55608288057, 1641837803793, 49218744365683, 1494112796918051, 45836491198618821, 1418839143493455861, 44259772786526485527, 1389967891240928450511, 43910122539568806384513, 1394423517592589134138485
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^n)^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + 9*x + 171*x^2 + 3819*x^3 + 94221*x^4 + 2474541*x^5 + 67842255*x^6 + 1919233719*x^7 + 55608288057*x^8 + 1641837803793*x^9 + 49218744365683*x^10 + ...

Crossrefs

Cf. A355865, A358952, A358953, A358954, A358955, A358956, A358957, A358958.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1,n, A=concat(A,0);
A[#A] = polcoeff( sum(n=-#A,#A, x^(9*n) * (x^n - 2*Ser(A))^(10*n+1) ), #A-1)/2);A[n+1]}
for(n=0,25,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
(1) 0 = Sum_{n=-oo..+oo} x^(9*n) * (x^n - 2*A(x))^(10*n+1).
(2) 0 = Sum_{n=-oo..+oo} x^(10*n*(n-1)) / (1 - 2*A(x)*x^n)^(10*n-1).

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

Views

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 + ...

Links

Crossrefs

Cf. A361771, A361773, A361774.
Cf. A363112, A355865, A357227, A359712, A357232.

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 + ...

Links

Crossrefs

Cf. A361771, A361772, A361774.
Cf. A363113, A355865, A357227, A359712, A357232.

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).

A358937 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (x^(2*n) - A(x))^n.

Original entry on oeis.org

1, 1, 3, 6, 13, 31, 76, 192, 504, 1351, 3668, 10082, 27991, 78335, 220778, 626141, 1785593, 5117179, 14729826, 42568767, 123465517, 359268141, 1048541699, 3068583485, 9002849260, 26474484680, 78019959584, 230381635121, 681544367457, 2019718168994, 5995000501189
Offset: 0

Views

Author

Paul D. Hanna, Dec 07 2022

Keywords

Comments

Related identity: 0 = Sum_{n=-oo..+oo} x^n * (y - x^(2*n+1))^n, which holds formally for all y.

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 6*x^3 + 13*x^4 + 31*x^5 + 76*x^6 + 192*x^7 + 504*x^8 + 1351*x^9 + 3668*x^10 + 10082*x^11 + 27991*x^12 + ...

Links

Crossrefs

Cf. A366229, A355865.

Programs

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

Formula

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) 1 = Sum_{n=-oo..+oo} x^n * (x^(2*n) - A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(n*(2*n-1)) / (1 - x^(2*n)*A(x))^n.
(3) 0 = Sum_{n=-oo..+oo} (-1)^n * x^n * (x^(2*n+1) - A(x))^n (trivial).
(4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 - x^(2*n)*A(x))^n (trivial).

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 + ...

Links

Crossrefs

Cf. A361772, A361773, A361774.
Cf. A355865, A357227, A359712, A357232.

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).

A361774 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, 150, 7003, 380817, 22517717, 1405927141, 91215539609, 6089092570148, 415519886498886, 28855638743197866, 2032628861705203315, 144884697917577076857, 10430845410431559928714, 757390467820895322043476, 55401570124877193188443429, 4078685155312165112343519832
Offset: 0

Views

Author

Paul D. Hanna, May 13 2023

Keywords

Examples

			G.f.: A(x) = 1 + 4*x + 150*x^2 + 7003*x^3 + 380817*x^4 + 22517717*x^5 + 1405927141*x^6 + 91215539609*x^7 + 6089092570148*x^8 + ...

Crossrefs

Cf. A361771, A361772, A361773.
Cf. A363114, A355865, A357227, A359712, A357232.

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)^(4*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)^(4*n-1).
(2) 1 = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(4*n^2) / (1 - 2*A(x)*(-x)^n)^(4*n+1).

A361778 Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * ((-x)^(n-1) - 2*A(x))^n.

Original entry on oeis.org

1, 2, 7, 27, 109, 459, 2006, 9017, 41384, 193048, 912571, 4361939, 21045710, 102361864, 501349447, 2470556294, 12240270901, 60935582862, 304660949343, 1529125824203, 7701783889261, 38915600049447, 197206343307012, 1002023916642621, 5103911800972155, 26056404563941575
Offset: 0

Views

Author

Paul D. Hanna, May 10 2023

Keywords

Examples

			G.f.: A(x) = 1 + 2*x + 7*x^2 + 27*x^3 + 109*x^4 + 459*x^5 + 2006*x^6 + 9017*x^7 + 41384*x^8 + 193048*x^9 + 912571*x^10 + ...
SPECIFIC VALUES.
A(1/7) = 1.63053651133635034184414884744745628155427916612173429157...
A(1/6) = 1.99892384479086071017436459041327119822244448085100733509...
A(x) = 2 at x = 0.166713109990638926829644490786806133084979604287174064...
Radius of convergence r and the value A(r) are given by
r = 0.182033752413024354859591633469061831146023401652842514076551...
A(r) = 2.63999965897091399750291467200041973752650665197493948118984006...
1/r = 5.4934867119096473651972990947886642212447897087082048838...

Links

Crossrefs

Cf. A355865, A357227, A359712.

Programs

  • PARI
    {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, x^m * ((-x)^(m-1) - 2*Ser(A))^m ), #A)/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 * ((-x)^(n-1) - 2*A(x))^n.
(2) 1 = Sum_{n=-oo..+oo} x^(2*n+1) * (2*A(x) - (-x)^n)^n.
(3) 2*A(x) = Sum_{n=-oo..+oo} x^(3*n+1) * ((-x)^n - 2*A(x))^n.
(4) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 - 2*A(x)*(-x)^(n+1))^n.
(5) 1 = Sum_{n=-oo..+oo} x^(n^2) / (1 - 2*A(x)*(-x)^(n+1))^(n+1).
(6) 2*A(x) = Sum_{n=-oo..+oo} (-1)^(n+1) * x^(n*(n-1)) / (1 - 2*A(x)*(-x)^(n+1))^(n+1).
(7) 0 = Sum_{n=-oo..+oo} x^(2*n) * (2*A(x) - (-x)^n)^(n+1).
(8) 0 = Sum_{n=-oo..+oo} x^(3*n) * ((-x)^(n-1) - 2*A(x))^(n+1).
Previous Showing 11-17 of 17 results.

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