A206592 E.g.f.: Sum_{n>=0} x^(n^2) * exp(n*x).

1, 1, 2, 3, 28, 245, 1446, 6727, 26888, 459657, 11208970, 180639371, 2158548492, 21024981133, 176560640270, 1324087390095, 30001965127696, 1480628781891857, 51566262458549778, 1299527188916481811, 25961751751545031700, 436032724081792884501

Offset

0,3

Comments

Compare to the partial theta series identity:

Sum_{n>=0} x^(n^2) = Sum_{n>=0} x^n * Product_{k=1..n} (1-x^(4*k-3))/(1-x^(4*k-1)).

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

Formula

E.g.f.: Sum_{n>=0} x^n*exp(n*x) * Product_{k=1..n} (1 - x^(4*k-3)*exp(x)) / (1 - x^(4*k-1)*exp(x)), due to a q-series identity.

The e.g.f. equals the continued fraction:

A(x) = 1/(1- x*exp(x)/(1- x*(x^2-1)*exp(x)/(1- x^5*exp(x)/(1- x^3*(x^4-1)*exp(x)/(1- x^9*exp(x)/(1- x^5*(x^6-1)*exp(x)/(1- x^13*exp(x)/(1- x^7*(x^8-1)*exp(x)/(1- ...))))))))), due to a partial elliptic theta function identity.

Example

G.f.: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! + 28*x^4/4! + 245*x^5/5! +...
where the e.g.f. is defined by:
A(x) = 1 + x*exp(x) + x^4*exp(2*x) + x^9*exp(3*x) + x^16*exp(4*x) +...
By a q-series identity:
A(x) = 1 + x*exp(x)*(1-x*exp(x))/(1-x^3*exp(x)) + x^2*exp(2*x)*(1-x*exp(x))*(1-x^5*exp(x))/((1-x^3*exp(x))*(1-x^7*exp(x))) + x^3*exp(3*x)*(1-x*exp(x))*(1-x^5*exp(x))*(1-x^9*exp(x))/((1-x^3*exp(x))*(1-x^7*exp(x))*(1-x^11*exp(x))) +...

Prog

(PARI) {a(n)=n!*polcoeff(sum(m=0, n+1, x^(m^2)*exp(m*x+x*O(x^n))), n)}
(PARI) {a(n)=local(A=1+x, X=x+x*O(x^n)); A=sum(m=0, n, x^m*exp(m*X)*prod(k=1, m, (1-x^(4*k-3)*exp(X))/(1-x^(4*k-1)*exp(X)))); n!*polcoeff(A, n)}
for(n=0, 35, print1(a(n), ", "))

Crossrefs

Cf. A193421, A206591.

Sequence in context: A354611 A356906 A371115 * A126266 A219975 A319146

Adjacent sequences: A206589 A206590 A206591 * A206593 A206594 A206595

Keyword

nonn

Author

Paul D. Hanna, Feb 09 2012

Status

approved