login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A206592
E.g.f.: Sum_{n>=0} x^(n^2) * exp(n*x).
2
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
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
Sequence in context: A354611 A356906 A371115 * A126266 A219975 A319146
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 09 2012
STATUS
approved