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A387288

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

1, 0, 2, 6, 10, 40, 110, 294, 938, 2688, 7932, 24090, 71434, 215072, 650078, 1958866, 5936970, 18012384, 54694100, 166470324, 507142300, 1546756560, 4723459622, 14436872714, 44166199466, 135231887440, 414370251700, 1270610340090, 3898731611610, 11970124471200

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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)^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 * (1 + x)^2) ). See A389062.

MATHEMATICA

Table[Sum[Binomial[n, k]*Binomial[2*k, n-2*k], {k, 0, Floor[n/2]}], {n, 0, 40}] (* Vincenzo Librandi, Sep 25 2025 *)

PROG

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