A388915 a(n) = Sum_{k=0..n} 3^k * binomial(n,k) * binomial(2*n+2*k,k).

1, 13, 289, 7228, 190417, 5166343, 142859224, 4003101661, 113276700433, 3229655233285, 92632940372299, 2669817506480137, 77257927623137320, 2243231587685499736, 65322014637963949021, 1906901680429247130928, 55788150746466567522769, 1635261226949615820226165

Offset

0,2

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..675

Formula

a(n) = [x^n] ((1+x)^2 * (x+3*(1+x)^2))^n.

The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/(3*x)) * Series_Reversion( x / ((1+x)^2 * (x+3*(1+x)^2)) ). See A388916.

Mathematica

Table[Sum[ 3^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, 3^k*binomial(n, k)*binomial(2*n+2*k, k));
(Magma) [&+[3^k*Binomial(n, k)*Binomial(2*n+2*k, k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Sep 24 2025

Crossrefs

Cf. A388916.

Sequence in context: A064750 A228753 A249864 * A246462 A375174 A023357

Adjacent sequences: A388912 A388913 A388914 * A388916 A388917 A388918

Keyword

nonn

Author

Seiichi Manyama, Sep 21 2025

Status

approved