A388915 - OEIS

%I #12 Sep 24 2025 10:19:54

%S 1,13,289,7228,190417,5166343,142859224,4003101661,113276700433,

%T 3229655233285,92632940372299,2669817506480137,77257927623137320,

%U 2243231587685499736,65322014637963949021,1906901680429247130928,55788150746466567522769,1635261226949615820226165

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

%H Vincenzo Librandi, <a href="/A388915/b388915.txt">Table of n, a(n) for n = 0..675</a>

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

%F 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.

%t Table[Sum[ 3^k* Binomial[ n,k]*Binomial[2*n+2*k,k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 24 2025 *)

%o (PARI) a(n) = sum(k=0, n, 3^k*binomial(n, k)*binomial(2*n+2*k, k));

%o (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

%Y Cf. A388916.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 21 2025