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A389151

a(n) = Sum_{k=0..floor(n/2)} binomial(n,k) * binomial(4*k,n-2*k).

1, 0, 2, 12, 30, 100, 446, 1596, 5726, 22512, 86112, 325160, 1255650, 4857996, 18728138, 72584512, 282194238, 1097448192, 4275805388, 16690807944, 65220896440, 255142508976, 999300655080, 3917625018604, 15371830602770, 60366665214000, 237247289826640, 933057346479960

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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^2 * (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^2 * (1 + x)^4) ). See A389155.

MATHEMATICA

Table[Sum[Binomial[n, k]*Binomial[4*k, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 30}] ( * Vincenzo Librandi, Sep 25 2025 *)

PROG

(PARI) a(n) = sum(k=0, n\2, binomial(n, k)*binomial(4*k, n-2*k));