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Self-convolution of A014070: a(n) = Sum_{k=0..n} C(2^k,k)*C(2^(n-k),n-k).
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%I #9 Jul 02 2016 08:00:16

%S 1,4,16,136,3900,410704,150779216,189354108224,819706419291728,

%T 12417873698752685696,668556572391910046409088,

%U 129665687275486846550512590336,91623983383737723477835280780455168,238057598315149125515904595621291745671168,2291332225550784443587332334013451028612830795776

%N Self-convolution of A014070: a(n) = Sum_{k=0..n} C(2^k,k)*C(2^(n-k),n-k).

%F G.f.: A(x) = Sum_{n>=0} (1/n!)*Sum_{k=0..n} C(n,k) * log(1+2^k*x)^k * log(1+2^(n-k)*x)^(n-k).

%F a(n) ~ 2^(n^2+1) / n!. - _Vaclav Kotesovec_, Jul 02 2016

%t Table[Sum[Binomial[2^k,k]*Binomial[2^(n-k),n-k], {k, 0, n}], {n, 0, 15}] (* _Vaclav Kotesovec_, Jul 02 2016 *)

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

%o for(n=0,20, print1(a(n),", "))

%o (PARI) {a(n) = polcoeff( sum(m=0,n, sum(k=0,m, log(1+2^k*x +x*O(x^n))^k/k! * log(1+2^(m-k)*x +x*O(x^n))^(m-k) / (m-k)! ) ),n)}

%o for(n=0,20, print1(a(n),", "))

%Y Cf. A014070 (C(2^n, n)).

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jan 16 2008