A389152 - OEIS

A389152

a(n) = Sum_{k=0..floor(n/3)} binomial(n,k) * binomial(4*k,n-3*k).

1, 0, 0, 3, 16, 30, 39, 175, 784, 2100, 4590, 13970, 50743, 153634, 408499, 1187823, 3843280, 11908160, 34470084, 101394222, 313440606, 967143586, 2897603170, 8660763804, 26408966551, 80995131580, 245801163530, 742776778965, 2260469097315, 6912033833250, 21074993569539

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OFFSET

0,4

LINKS

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

FORMULA

a(n) = [x^n] (1 + x^3 * (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^3 * (1 + x)^4) ). See A389156.

MATHEMATICA

Table[Sum[Binomial[n, k]Binomial[4*k, n-3*k], {k, 0, Floor[n/3]}], {n, 0, 40}] (* Vincenzo Librandi, Sep 26 2025 *)

PROG

(PARI) a(n) = sum(k=0, n\3, binomial(n, k)*binomial(4*k, n-3*k));