A251671 a(n) = Sum_{k=0..n} C(n,k) * (2^k + 3^k)^k.

1, 6, 180, 43398, 88701816, 1573206748746, 248688444559874580, 356335498302585834118638, 4663871943514788530035646937456, 558720685051192771669885091319459750546, 612058892657175926094223171960469926874935754700

Offset: 0

Author

Paul D. Hanna, Jan 21 2015

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + 6*x + 180*x^2 + 43398*x^3 + 88701816*x^4 + 1573206748746*x^5 +...
where A(x) = 1/(1-x) + (2+3)*x/(1-x)^2 + (2^2+3^2)*x^2/(1-x)^3 + (2^3+3^3)^3*x^3/(1-x)^4 +...
ILLUSTRATION OF INITIAL TERMS:
a(0) = 1*(2^0+3^0)^0 = 1;
a(1) = 1*(2^0+3^0)^0 + 1*(2^1+3^1)^1 = 6;
a(2) = 1*(2^0+3^0)^0 + 2*(2^1+3^1)^1 + 1*(2^2+3^2)^2 = 180;
a(3) = 1*(2^0+3^0)^0 + 3*(2^1+3^1)^1 + 3*(2^2+3^2)^2 + 1*(2^3+3^3)^3 = 43398;
a(4) = 1*(2^0+3^0)^0 + 4*(2^1+3^1)^1 + 6*(2^2+3^2)^2 + 4*(2^3+3^3)^3 + 1*(2^4+3^4)^4 = 88701816; ...
		

Crossrefs

Cf. A251661, A251184, A060722.

Programs

Mathematica

Table[Sum[Binomial[n,k] * (2^k + 3^k)^k,{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Jan 25 2015 *)

PARI

{a(n)=sum(k=0, n, binomial(n, k) * (2^k + 3^k)^k )}
for(n=0, 15, print1(a(n), ", "))

PARI

{a(n)=local(A=1); A=sum(m=0, n, (2^m + 3^m)^m * x^m / (1-x +x*O(x^n) )^(m+1) ); polcoeff(A, n)}
for(n=0, 15, print1(a(n), ", "))

Formula

G.f.: Sum_{n>=0} (2^n + 3^n)^n * x^n / (1-x)^(n+1).

a(n) ~ 3^(n^2). - Vaclav Kotesovec, Jan 25 2015