A387432 a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*n+3*k,k).

1, 8, 114, 1835, 31138, 544338, 9699711, 175165474, 3194598394, 58704371324, 1085247078064, 20160490831874, 376031731326271, 7037553889905222, 132092061196259124, 2485522166002805295, 46871155550986528714, 885578304615102454136, 16760505412564169454024

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..600

Formula

a(n) = [x^n] ((1+x)^4 * (x+(1+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+x)^4 * (x+(1+x)^3)) ). See A389273.

a(n) = hypergeom([-n, (1+4*n)/3, (2+4*n)/3, 1+4*n/3], [1, 1/2+2*n, 1+2*n], -3^3/2^2)/(1 + n). - Stefano Spezia, Sep 28 2025

Mathematica

Table[Sum[Binomial[n, k]Binomial[4*n+3*k, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Oct 08 2025 *)

Prog

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(4*n+3*k, k));
(Magma) [&+[Binomial(n, k) * Binomial(4*n+3*k, k) : k in [0..n] ]: n in [0..40]]; // Vincenzo Librandi, Oct 08 2025

Crossrefs

Cf. A156887, A388729.

Cf. A389273.

Sequence in context: A062126 A222520 A220635 * A386763 A113670 A347019

Adjacent sequences: A387429 A387430 A387431 * A387433 A387434 A387435

Keyword

nonn

Author

Seiichi Manyama, Sep 28 2025

Status

approved