A135754 E.g.f.: A(x) = Sum_{n>=0} exp((4^n-1)/3*x)*x^n/n!.
1, 1, 3, 19, 239, 6091, 305023, 30818299, 6155906879, 2484667187371, 1989929726352863, 3221489148102557179, 10362312712649347408159, 67345216546226371822133611, 869978904614825017953532433663
Offset: 0
Keywords
Links
- G. C. Greubel, Table of n, a(n) for n = 0..80
Programs
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Mathematica
Flatten[{1,Table[Sum[Binomial[n,k]*((4^k-1)/3)^(n-k),{k,0,n}],{n,1,20}]}] (* Vaclav Kotesovec, Jun 25 2013 *) -
PARI
a(n)=sum(k=0,n,binomial(n,k)*((4^k-1)/3)^(n-k))
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PARI
a(n)=n!*polcoeff(sum(k=0,n,exp((4^k-1)/3*x)*x^k/k!),n)
Formula
a(n) = Sum_{k=0..n} C(n,k)*[(4^k-1)/3]^(n-k).
a(n) ~ c * 2^(n^2/2+n+1/2)/(3^(n/2)*sqrt(Pi*n)), where c = Sum_{k = -infinity..infinity} 3^k*4^(-k^2) = 1.86902676808473931... if n is even and c = Sum_{k = -infinity..infinity} 3^(k+1/2)*4^(-(k+1/2)^2) = 1.87384213421283135... if n is odd. - Vaclav Kotesovec, Jun 25 2013