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

0,3

Links

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

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

Mathematica

Table[Sum[(-1)^(n-k)*Binomial[n, k]*Binomial[2^k, k], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)

Prog

(PARI) {a(n)=sum(k=0, n, (-1)^(n-k)*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)}

Crossrefs

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

Sequence in context: A303159 A274387 A300988 * A114337 A317343 A307248

Adjacent sequences: A136645 A136646 A136647 * A136649 A136650 A136651

Keyword

nonn

Author

Paul D. Hanna and Vladeta Jovovic, Jan 21 2008

Extensions

Edited by Charles R Greathouse IV, Oct 28 2009

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

Status

approved