A206592 - OEIS

%I #7 Sep 03 2017 10:40:21

%S 1,1,2,3,28,245,1446,6727,26888,459657,11208970,180639371,2158548492,

%T 21024981133,176560640270,1324087390095,30001965127696,

%U 1480628781891857,51566262458549778,1299527188916481811,25961751751545031700,436032724081792884501

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

%C Compare to the partial theta series identity:

%C 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)).

%H Vaclav Kotesovec, <a href="/A206592/b206592.txt">Table of n, a(n) for n = 0..440</a>

%F 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.

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

%F 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.

%e G.f.: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! + 28*x^4/4! + 245*x^5/5! +...

%e where the e.g.f. is defined by:

%e A(x) = 1 + x*exp(x) + x^4*exp(2*x) + x^9*exp(3*x) + x^16*exp(4*x) +...

%e By a q-series identity:

%e 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))) +...

%o (PARI) {a(n)=n!*polcoeff(sum(m=0,n+1,x^(m^2)*exp(m*x+x*O(x^n))),n)}

%o (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)}

%o for(n=0, 35, print1(a(n), ", "))

%Y Cf. A193421, A206591.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 09 2012