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 1-3 of 3 results.

A384950 a(n) = Sum_{k=0..n} 3^k * binomial(2*n+k-1,k).

Original entry on oeis.org

1, 7, 103, 1720, 30319, 550867, 10204660, 191606380, 3633593071, 69434167357, 1334845289023, 25787841299392, 500217562201348, 9736067678711524, 190051513661403112, 3719197868485767940, 72942019051301120239, 1433317465944902210161, 28212929859612197439829
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A383888, A385438.
Cf. A385474.

Programs

  • PARI
    a(n) = sum(k=0, n, 3^k*binomial(2*n+k-1, k));

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k).
a(n) = [x^n] ( (1+x)^3/(1-2*x)^2 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+x)^3 ). See A385474.
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n,k).
a(n) = 4^(-n) - 3^n*binomial(3*n-1, n)*(hypergeom([1, 3*n], [1+n], 3) - 1). - Stefano Spezia, Aug 02 2025
a(n) ~ 3^(4*n + 3/2) / (2^(2*n+3) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 04 2025

A385475 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^3 / (1+x)^4 ).

Original entry on oeis.org

1, 10, 154, 2836, 57601, 1244584, 28063288, 652821724, 15551944804, 377503375150, 9303441938506, 232168129150420, 5854967533764766, 148981015820615968, 3820184959840942564, 98616983735455104412, 2560818171703792341484, 66845502538144505160040
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2025

Keywords

Links

Crossrefs

Cf. A064063, A385474.
Cf. A386723.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^3/(1+x)^4)/x)
  • PARI
    a(n) = sum(k=0, n, 2^(n-k)*binomial(4*(n+1), k)*binomial(4*n-k+2, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 2^(n-k) * binomial(4*(n+1),k) * binomial(4*n-k+2,n-k).
a(n) = (1/(n+1)) * [x^n] ( (1+x)^4 / (1-2*x)^3 )^(n+1).

A386770 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+3*x)^3 ).

Original entry on oeis.org

1, 13, 244, 5397, 130961, 3372268, 90497184, 2503434117, 70883043571, 2044268649573, 59842331451024, 1773506049794412, 53107658756034156, 1604418047921589928, 48841208603255888264, 1496711470907670605157, 46134317696761847385591, 1429405788411234205692583
Offset: 0

Views

Author

Seiichi Manyama, Aug 02 2025

Keywords

Links

Crossrefs

Cf. A385474, A386764.

Programs

  • PARI
    my(N=20, x='x+O('x^N)); Vec(serreverse(x*(1-2*x)^2/(1+3*x)^3)/x)
  • PARI
    a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(3*(n+1), k)*binomial(3*n-k+1, n-k))/(n+1);

Formula

a(n) = (1/(n+1)) * Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*(n+1),k) * binomial(3*n-k+1,n-k).
a(n) = (1/(n+1)) * [x^n] ( (1+3*x)^3 / (1-2*x)^2 )^(n+1).
D-finite with recurrence 32*(n+1)*(2*n+1)*a(n) +48*(-81*n^2+27*n-7)*a(n-1) +162*(414*n^2-891*n+605)*a(n-2) -32805*(3*n-4)*(3*n-5)*a(n-3)=0. - R. J. Mathar, Aug 03 2025
Showing 1-3 of 3 results.

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

Content is available under The OEIS End-User License Agreement.