A168481 G.f.: Sum_{n>=0} (n+1)*2^(n^2)*(1 + 2^n*x)^n*x^n.

1, 4, 56, 2432, 377600, 222691328, 513752956928, 4690384533848064, 170085542794237050880, 24520078828632712041988096, 14055876186039467842015007342592

Offset

0,2

Comments

This sequence illustrates the identity:

Sum_{n>=0} (n+1)*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))^2.

Links

Table of n, a(n) for n=0..10.

Formula

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

a(n) = Sum_{k=0..[n/2]} (n-k+1)*C(n-k,k)*2^(n(n-k)).

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

Example

G.f.: A(x) = 1 + 4*x + 56*x^2 + 2432*x^3 + 377600*x^4 +...

Mathematica

Table[Sum[(n-k+1)*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, (m+1)*(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))^2, n)}
(PARI) {a(n)=sum(k=0, n\2, (n-k+1)*binomial(n-k, k)*2^(n*(n-k)))}

Crossrefs

Cf. A168480, A168482.

Sequence in context: A012959 A013113 A009105 * A171801 A091797 A265230

Adjacent sequences: A168478 A168479 A168480 * A168482 A168483 A168484

Keyword

nonn

Author

Paul D. Hanna, Nov 26 2009

Status

approved