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A136649 Binomial transform of A014070: a(n) = Sum_{k=0..n} C(n,k)*C(2^k,k).

%I A136649 #17 Jun 26 2022 09:17:18
%S A136649 1,3,11,81,2089,211107,76211147,95054910473,410422012327681,
%T A136649 6211807332775516467,334327967114349983723899,
%U A136649 64835852334793138873642165105,45812640033676518721399820389451657
%N A136649 Binomial transform of A014070: a(n) = Sum_{k=0..n} C(n,k)*C(2^k,k).
%H A136649 G. C. Greubel, <a href="/A136649/b136649.txt">Table of n, a(n) for n = 0..59</a>
%F A136649 G.f.: A(x) = (1/(1-x))*Sum_{n>=0} [log(1 + (2^n-1)*x) - log(1-x)]^n / n!.
%F A136649 From _Vaclav Kotesovec_, Jul 02 2016: (Start)
%F A136649 a(n) ~ binomial(2^n,n).
%F A136649 a(n) ~ 2^(n^2) / n!.
%F A136649 a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)). (End)
%t A136649 Table[Sum[Binomial[n,k]*Binomial[2^k,k],{k,0,n}],{n,0,15}] (* _Vaclav Kotesovec_, Jul 02 2016 *)
%o A136649 (PARI) {a(n)=sum(k=0,n,binomial(n,k)*binomial(2^k,k))}
%o A136649 (PARI) /* Using the g.f.: */ {a(n)=local(X=x+x*O(x^n));polcoeff(sum(k=0,n,(log(1+(2^k-1)*X)-log(1-X))^k/k!)/(1-X),n)}
%Y A136649 Cf. A014070 (C(2^n, n)), A134173.
%Y A136649 Partial sums of A180687.
%K A136649 nonn
%O A136649 0,2
%A A136649 _Paul D. Hanna_ and _Vladeta Jovovic_, Jan 21 2008
%E A136649 Edited by _Charles R Greathouse IV_, Oct 28 2009