A168482 G.f.: Sum_{n>=0} [(n+1)(n+2)/2]*2^(n^2)*(1 + 2^n*x)^n*x^n.

1, 6, 108, 5888, 1107456, 768540672, 2038400811008, 21012301788217344, 848380447466572480512, 134688875052459668084359168, 84279244669071810947388769566720

Offset

0,2

Comments

This sequence illustrates the identity:

Sum_{n>=0} [(n+1)(n+2)/2]*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))^3.

Links

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

Formula

a(n) = [x^n] 1/(1 - 2^n*x*(1+x))^3.

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

a(n) ~ n^2 * 2^(n^2 - 1). - Vaclav Kotesovec, Oct 07 2020

Example

G.f.: A(x) = 1 + 6*x + 108*x^2 + 5888*x^3 + 1107456*x^4 +...

Mathematica

Table[Sum[(n - k + 1)*(n - k + 2)/2 * Binomial[n - k, k]*2^(n*(n - k)), {k, 0, n/2}], {n, 0, 15}] (* Vaclav Kotesovec, Oct 07 2020 *)

Prog

(PARI) {a(n)=polcoeff(sum(m=0, n, (m+1)*(m+2)/2*(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))^3, n)}
(PARI) {a(n)=sum(k=0, n\2, (n-k+1)*(n-k+2)/2*binomial(n-k, k)*2^(n*(n-k)))}

Crossrefs

Cf. A168480, A168481.

Sequence in context: A122722 A127946 A012503 * A132856 A368851 A337580

Adjacent sequences: A168479 A168480 A168481 * A168483 A168484 A168485

Keyword

nonn

Author

Paul D. Hanna, Nov 26 2009

Status

approved