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.

A135863 G.f. A(x) = 1 + 4*x*A(x)^(1/2); A(x) = 1 + 8*x^2 + 4*x*sqrt(1 + 4*x^2).

Original entry on oeis.org

1, 4, 8, 8, 0, -8, 0, 16, 0, -40, 0, 112, 0, -336, 0, 1056, 0, -3432, 0, 11440, 0, -38896, 0, 134368, 0, -470288, 0, 1664096, 0, -5943200, 0, 21395520, 0, -77558760, 0, 282861360, 0, -1037158320, 0, 3821109600, 0, -14138105520, 0, 52512963360, 0, -195730136160
Offset: 0

Views

Author

Paul D. Hanna, Dec 02 2007

Keywords

Crossrefs

Cf. A135864, A214377, A000108.

Programs

  • PARI
    a(n)=4^n*if(n==2,1/2,binomial(n/2,n)/(1-n/2))

Formula

a(n) = -4^n*binomial(n/2,n)/(n/2 - 1), except a(2) = 8, for n>=0.
G.f.: (exp(asinh(2*x)))^2. - Philippe Deléham, Feb 01 2012
D-finite with recurrence: (-n+1)*a(n) +(-n+2)*a(n-1) +4*(-n+4)*a(n-2) +4*(-n+5)*a(n-3)=0. - R. J. Mathar, Jan 23 2020
From Alexander Burstein, Mar 27 2022: (Start)
G.f. satisfies: A(-x) = 1/A(x).
a(2*n+3) = (-1)^n*8*A000108(n) for n>=0. (End)

A385117 G.f. A(x) satisfies A(x) = 1 + 9*x*A(x)^(2/3).

Original entry on oeis.org

1, 9, 54, 243, 810, 1701, 0, -16038, -56862, 0, 817938, 3241134, 0, -53872371, -224386200, 0, 4017339666, 17216031195, 0, -322568743770, -1408090130370, 0, 27206369474544, 120309415164990, 0, -2376712950727284, -10611290417552118, 0, 213172869272924088
Offset: 0

Views

Author

Seiichi Manyama, Jun 18 2025

Keywords

Links

Crossrefs

Cf. A135864, A214668, A385119.
Cf. A376636.

Programs

  • Mathematica
    A385117[n_] := 9^n*Binomial[2*n/3 + 1, n]/(2*n/3 + 1);
Array[A385117, 35, 0] (* Paolo Xausa, Aug 01 2025 *)
  • PARI
    a(n) = 9^n*binomial(2*n/3+1, n)/(2*n/3+1);

Formula

a(n) = 9^n * binomial(2*n/3+1,n)/(2*n/3+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^(1/3)).
G.f.: 1/B(-x), where B(x) is the g.f. of A135864.
G.f.: B(x)^3, where B(x) is the g.f. of A376636.
a(3*n) = 0 for n > 1.
D-finite with recurrence (n-1)*(n-2)*a(n) + 54*(2*n-3)*(n-6)*a(n-3) = 0. - R. J. Mathar, Jul 30 2025
a(n) ~ A128834(n) * 2^(2*n/3) * 3^(n+3/2) / (sqrt(Pi) * n^(3/2)). - Amiram Eldar, Sep 02 2025

A385119 G.f. A(x) satisfies A(x) = 1 + 9*x*A(x)^(5/3).

Original entry on oeis.org

1, 9, 135, 2430, 48195, 1015740, 22320522, 505692720, 11727186075, 277005649635, 6641224015140, 161193712078854, 3953072078945730, 97801207953712200, 2438092322304120720, 61182608813245896840, 1544295394480280288715, 39180450803555268621540
Offset: 0

Views

Author

Seiichi Manyama, Jun 18 2025

Keywords

Links

Crossrefs

Cf. A135864, A214668, A385117.
Cf. A245114.

Programs

  • Mathematica
    A385119[n_] := 9^n*Binomial[#, n]/# & [5*n/3 + 1];
Array[A385119, 20, 0] (* Paolo Xausa, Aug 05 2025 *)
  • PARI
    a(n) = 9^n*binomial(5*n/3+1, n)/(5*n/3+1);

Formula

a(n) = 9^n * binomial(5*n/3+1,n)/(5*n/3+1).
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^(7/3)).
G.f.: B(x)^3, where B(x) is the g.f. of A245114.
D-finite with recurrence 2*n*(n-1)*(n-2)*(2*n+3)*a(n) - 135*(5*n-9)*(5*n-3)*(5*n-12)*(5*n-6)*a(n-3) = 0. - R. J. Mathar, Jul 30 2025
a(n) ~ 3^(n+1) * 5^(5*n/3+1/2) / (sqrt(Pi) * 2^(2*(n+3)/3) * n^(3/2)). - Amiram Eldar, Sep 02 2025

A135866 G.f. A(x) satisfies: A(x) = 1 + x*A(9x)^(1/3).

Original entry on oeis.org

1, 1, 3, 72, 17055, 37191123, 731807102304, 129633035920625844, 206676152816612330273175, 2965575655806580962693168234624, 382974905786208201171725242242120141033
Offset: 0

Views

Author

Paul D. Hanna, Dec 02 2007

Keywords

Crossrefs

Cf. A135863, A135864.

Programs

  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=1+x*subst(A,x,9*x)^(1/3));polcoeff(A,n)}
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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