A136649 Binomial transform of A014070: a(n) = Sum_{k=0..n} C(n,k)*C(2^k,k).
1, 3, 11, 81, 2089, 211107, 76211147, 95054910473, 410422012327681, 6211807332775516467, 334327967114349983723899, 64835852334793138873642165105, 45812640033676518721399820389451657
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..59
Programs
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Mathematica
Table[Sum[Binomial[n,k]*Binomial[2^k,k],{k,0,n}],{n,0,15}] (* Vaclav Kotesovec, Jul 02 2016 *) -
PARI
{a(n)=sum(k=0,n,binomial(n,k)*binomial(2^k,k))} -
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)}
Formula
G.f.: A(x) = (1/(1-x))*Sum_{n>=0} [log(1 + (2^n-1)*x) - log(1-x)]^n / n!.
From Vaclav Kotesovec, Jul 02 2016: (Start)
a(n) ~ binomial(2^n,n).
a(n) ~ 2^(n^2) / n!.
a(n) ~ 2^(n^2 - 1/2) * exp(n) / (sqrt(Pi) * n^(n+1/2)). (End)
Extensions
Edited by Charles R Greathouse IV, Oct 28 2009