OFFSET
1,2
COMMENTS
A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..401
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
FORMULA
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 4*A(x))^n = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 4*A(x))^(n-1) = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 4*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^n)^n = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^n)^(n+1) = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 4*A(x)*x^n)^(n+1) = 0.
EXAMPLE
G.f.: A(x) = x + 4*x^2 + 19*x^3 + 99*x^4 + 529*x^5 + 2853*x^6 + 15566*x^7 + 85879*x^8 + 477716*x^9 + 2674070*x^10 + 15047671*x^11 + 85063429*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 4*A(x))^n = 1 - 6*x + 6*x^4 - 6*x^9 + 6*x^16 - 6*x^25 + 6*x^36 - 6*x^49 +- ...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, (-1)^m * (x^m + 4*Ser(A))^m ) - 1 - 6*sum(m=1, #A, (-1)^m * x^(m^2) ), #A-1)/4 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 09 2024
STATUS
approved