A388912 - OEIS

A388912

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

1, 6, 56, 594, 6652, 76776, 903416, 10774308, 129775388, 1575057960, 19231572576, 235968734106, 2907035046820, 35935514538072, 445510834281744, 5537082937554504, 68968824909560604, 860715248856275880, 10759852369305887648, 134714333266052865960, 1688941704992366810992

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OFFSET

0,2

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..800

FORMULA

a(n) = [x^n] ((1+x)^2 * (2*x+(1+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+x)^2 * (2*x+(1+x)^2)) ). See A388914.

MATHEMATICA

Table[Sum[ 2^(n-k)* Binomial[ n, k]*Binomial[2*n+2*k, k], {k, 0, n}], {n, 0, 30}] (* Vincenzo Librandi, Sep 24 2025 *)

PROG

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