A168480 G.f.: Sum_{n>=0} 2^(n^2)*(1 + 2^n*x)^n*x^n.

1, 2, 20, 640, 78080, 37847040, 74189111296, 589682903613440, 18955380356036952064, 2455824622368881511497728, 1278825951842748707166092263424, 2671459568763422966186162922297753600

Offset

0,2

Comments

This sequence illustrates the identity:

Sum_{n>=0} q^(n^2)*G(q^n*x)^n*x^n = Sum_{n>=0} c(n)*x^n

where c(n) = [x^n] 1/(1 - q^n*x*G(x)).

Links

Table of n, a(n) for n=0..11.

Formula

a(n) = [x^n] 1/(1 - 2^n*x*(1+x)).

a(n) = Sum_{k=0..[n/2]} C(n-k,k)*2^(n(n-k)).

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Nov 05 2014

Example

G.f.: A(x) = 1 + 2*x + 20*x^2 + 640*x^3 + 78080*x^4 +...

Mathematica

Table[Sum[Binomial[n-k, k]*2^(n*(n-k)), {k, 0, Floor[n/2]}], {n, 0, 15}] (* Vaclav Kotesovec, Nov 05 2014 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, (1+2^m*x)^m*2^(m^2)*x^m)+x*O(x^n), n)}
(PARI) {a(n)=polcoeff(1/(1-2^n*x*(1+x)+x*O(x^n)), n)}
(PARI) {a(n)=sum(k=0, n\2, binomial(n-k, k)*2^(n*(n-k)))}

Crossrefs

Cf. A168481, A168482.

Sequence in context: A015207 A054941 A012495 * A364886 A198761 A171799

Adjacent sequences: A168477 A168478 A168479 * A168481 A168482 A168483

Keyword

nonn

Author

Paul D. Hanna, Nov 26 2009

Status

approved