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.

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A385438 a(n) = Sum_{k=0..n} 3^k * binomial(3*n+k-1,k).

Original entry on oeis.org

1, 10, 208, 4888, 121132, 3092950, 80506684, 2123780536, 56581885468, 1518936682888, 41021505946468, 1113273696074968, 30335161535834212, 829405495046080612, 22742967214283811976, 625193974445825554408, 17223870801864911429404, 475423918887141016417144
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2025

Keywords

Crossrefs

Programs

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

Formula

a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k).
a(n) = [x^n] ( (1+x)^4/(1-2*x)^3 )^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)^3 / (1+x)^4 ). See A385475.
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(4*n,k).
a(n) = (-2)^(-3*n) - 3^n*binomial(4*n-1, n)*(hypergeom([1, 4*n], [1+n], 3) - 1). - Stefano Spezia, Aug 02 2025
a(n) ~ 2^(8*n + 1/2) / (11 * 3^(2*n - 3/2) * sqrt(Pi*n)). - Vaclav Kotesovec, Aug 04 2025

A385474 Expansion of (1/x) * Series_Reversion( x * (1-2*x)^2 / (1+x)^3 ).

Original entry on oeis.org

1, 7, 76, 991, 14281, 219172, 3512440, 58096591, 984340003, 16996883887, 298017184048, 5291703108292, 94961611382860, 1719543577996888, 31379622840361696, 576519956457976495, 10655055147825932119, 197959348525977645781, 3695112941037246866044
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2025

Keywords

Crossrefs

Programs

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