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a(n) = Sum_{k=0..n} binomial(n,k) * 2^((n-k)^2) * 3^(k^2).
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%I #6 Sep 03 2017 07:38:14

%S 1,5,109,20825,43283641,847757178125,150104882696162149,

%T 239301431405467344190625,3433687649167507509801752071921,

%U 443426550049486796441016276819404703125,515377529600543569431994967945053326153797481949

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

%H Paul D. Hanna, <a href="/A245106/b245106.txt">Table of n, a(n) for n = 0..45</a>

%F E.g.f.: ( Sum_{n>=0} 2^(n^2)*x^n/n! ) * ( Sum_{n>=0} 3^(n^2)*x^n/n! ).

%F a(n) ~ 3^(n^2). - _Vaclav Kotesovec_, Sep 03 2017

%e E.g.f.: A(x) = 1 + 5*x + 109*x^2/2! + 20825*x^3/3! + 43283641*x^4/4! + 847757178125*x^5/5! +...

%e where A(x) = B(x)*C(x) with

%e B(x) = 1 + 2*x + 2^4*x^2/2! + 2^9*x^3/3! + 2^16*x^4/4! + 2^25*x^5/5! +...

%e C(x) = 1 + 3*x + 3^4*x^2/2! + 3^9*x^3/3! + 3^16*x^4/4! + 3^25*x^5/5! +...

%o (PARI) {a(n)=sum(k=0,n,binomial(n,k)*2^((n-k)^2)*3^(k^2))}

%o for(n=0,16,print1(a(n),", "))

%Y Cf. A197356.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jul 12 2014