A136648 Inverse binomial transform of A014070: a(n) = Sum_{k=0..n} (-1)^(n-k)*C(n,k)*C(2^k,k).

1, 1, 3, 43, 1625, 192785, 73792371, 94005141667, 408909577044065, 6204433373664395569, 334203804752658372354515, 64828498485572980097719939179, 45811084061472137471487315433296153, 119028111984311982345314987179033877373025, 1145664208319965667452046935744516601565935434531

Offset: 0

Author

Paul D. Hanna and Vladeta Jovovic, Jan 21 2008

Keywords

• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A014070 (C(2^n, n)), A134174.

Programs

Mathematica

Table[Sum[(-1)^(n-k)*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,(-1)^(n-k)*binomial(n,k)*binomial(2^k,k))}

PARI

/* Using the g.f.: */ {a(n)=my(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!.

a(n) ~ 2^(n^2) / n!. - Vaclav Kotesovec, Jul 02 2016

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009

Terms a(13) and beyond from Andrew Howroyd, Feb 02 2020