A206591 - OEIS

%I #10 Sep 03 2017 10:53:16

%S 1,1,2,3,28,485,5766,53767,430088,3459465,53303050,1746391691,

%T 58977262092,1706810202253,42923448632078,965348202349455,

%U 19877420584519696,385436337079476497,7654870637722391058,199927590326456092435,8556099311090244142100

%N E.g.f.: Sum_{n>=0} x^(n^2)*exp(n^2*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="/A206591/b206591.txt">Table of n, a(n) for n = 0..400</a>

%F E.g.f.: Sum_{n>=0} x^n*exp(n*x) * Product_{k=1..n} (1 - x^(4*k-3)*exp((4*k-3)*x))/(1 - x^(4*k-1)*exp((4*k-1)*x)).

%F Let q = x*exp(x), then the e.g.f. equals the continued fraction:

%F A(x) = 1/(1- q/(1- q*(q^2-1)/(1- q^5/(1- q^3*(q^4-1)/(1- q^9/(1- q^5*(q^6-1)/(1- q^13/(1- q^7*(q^8-1)/(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! + 485*x^5/5! +...

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

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

%e Let q = x*exp(x), then the e.g.f. also equals the q-series:

%e A(x) = 1 + q*(1-q)/(1-q^3) + q^2*(1-q)*(1-q^5)/((1-q^3)*(1-q^7)) + q^3*(1-q)*(1-q^5)*(1-q^9)/((1-q^3)*(1-q^7)*(1-q^11)) +...

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

%o (PARI) {a(n)=local(X=x+x*O(x^n));n!*polcoeff(1+sum(m=1, n+1, x^m*exp(m*X)*prod(k=1, m,(1 - x^(4*k-3)*exp((4*k-3)*X))/(1 - x^(4*k-1)*exp((4*k-1)*X))) ), n)}

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

%Y Cf. A193421, A206592.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Feb 09 2012