A387931 - OEIS

%I #18 Sep 21 2025 10:34:50

%S 1,6,60,678,8076,99096,1239396,15709140,201078396,2593341768,

%T 33647629680,438693096870,5742698276340,75429886563336,

%U 993636177253080,13121886019508808,173664637285538268,2302815409326783720,30587580089238772176,406902359573509985784,5420349836578927206576

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

%H Vincenzo Librandi, <a href="/A387931/b387931.txt">Table of n, a(n) for n = 0..600</a>

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

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

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

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

%F a(n) ~ 2^(2*n - 1/2) * (130*sqrt(10) + 419)^n / (sqrt((7*sqrt(10) - 20)*Pi*n) * 3^(5*n - 1/2)). - _Vaclav Kotesovec_, Sep 21 2025

%t Table[Sum[ 3^(n-k)*Binomial[ n,k]*Binomial[3*n,k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 20 2025 *)

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

%o (Magma) [&+[3^(n-k)*Binomial(n, k)*Binomial(3*n,k): k in [0..n]]: n in [0..20]]; // _Vincenzo Librandi_, Sep 20 2025

%Y Cf. A387929, A387933.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Sep 13 2025