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 + 3*A(x))^n = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 3*A(x))^(n-1) = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 3*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^n)^n = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 3*A(x)*x^n)^(n+1) = 1 + 5*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 3*A(x)*x^n)^(n+1) = 0.
EXAMPLE
G.f.: A(x) = x + 3*x^2 + 12*x^3 + 53*x^4 + 234*x^5 + 1041*x^6 + 4711*x^7 + 21573*x^8 + 99484*x^9 + 461657*x^10 + 2154591*x^11 + 10102701*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 3*A(x))^n = 1 - 5*x + 5*x^4 - 5*x^9 + 5*x^16 - 5*x^25 + 5*x^36 - 5*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05001316702398359971645418498866690386932728399152644693...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 3*A)^n = (5*(Pi/2)^(1/4)/gamma(3/4) - 3)/2 = 0.78394784539029205351810...
(V.2) Let A = A(exp(-2*Pi)) = 0.001877983557643657576778844718492775838546798118866577860...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 3*A)^n = (5*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 3)/2 = 0.990662786402267839474...
(V.3) Let A = A(-exp(-Pi)) = -0.03842474691590612761867206263978602696713545771404819339...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 3*A)^n = (5*Pi^(1/4)/gamma(3/4) - 3)/2 = 1.216087028033270036438...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001857058214293085256892081751882664927312970576990961749...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 3*A)^n = (5*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 3)/2 = 1.009337213719347727619...
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 + 3*Ser(A))^m ) - 1 - 5*sum(m=1, #A, (-1)^m * x^(m^2) ), #A-1)/3 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 09 2024
STATUS
approved