A389225 - OEIS

A389225

a(n) = Sum_{k=0..n} binomial(n,k) * binomial(4*k,n-k).

1, 1, 9, 43, 233, 1306, 7371, 42260, 244329, 1422655, 8331034, 49011084, 289442363, 1714910250, 10188959132, 60682788448, 362173743273, 2165588486477, 12970369027287, 77798339136333, 467267412853338, 2809846878501083, 16915084390269668, 101929205246326797

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OFFSET

0,3

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

FORMULA

a(n) = [x^n] (1 + x * (1 + x)^4)^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 * (1 + x)^4) ). See A364743.

MATHEMATICA

Table[Sum[Binomial[n, k]Binomial[4*k, n-k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Oct 09 2025 *)

PROG

(PARI) a(n) = sum(k=0, n, binomial(n, k)*binomial(4*k, n-k));