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 + 6*A(x))^n = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 6*A(x))^(n-1) = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 6*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 6*A(x)*x^n)^n = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 6*A(x)*x^n)^(n+1) = 1 + 8*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 6*A(x)*x^n)^(n+1) = 0.
EXAMPLE
G.f.: A(x) = x + 6*x^2 + 39*x^3 + 269*x^4 + 1917*x^5 + 13893*x^6 + 101830*x^7 + 753255*x^8 + 5614504*x^9 + 42110432*x^10 + 317474187*x^11 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 6*A(x))^n = 1 - 8*x + 8*x^4 - 8*x^9 + 8*x^16 - 8*x^25 + 8*x^36 - 8*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05892551210473733684254468528377030200762221986684224912...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 6*A)^n = 4*(Pi/2)^(1/4)/gamma(3/4) - 3 = 0.65431655262446728562897...
(V.2) Let A = A(exp(-2*Pi)) = 0.001888624085511713374935799800784148455986111369097248489...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 6*A)^n = 4*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 3 = 0.985060458243628543159...
(V.3) Let A = A(-exp(-Pi)) = -0.03443859231795915470687740421610270983167641847531807729...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 6*A)^n = 4*Pi^(1/4)/gamma(3/4) - 3 = 1.3457392448532320583012...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001846769433141026637620872576636896819075507182864480219...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 6*A)^n = 4*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 3 = 1.01493954195095636419...
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 + 6*Ser(A))^m ) - 1 - 8*sum(m=1, #A, (-1)^m * x^(m^2) ), #A-1)/6 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 09 2024
STATUS
approved