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 + 2*A(x))^n = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 2*A(x))^(n-1) = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 2*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 2*A(x)*x^n)^n = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 2*A(x)*x^n)^(n+1) = 1 + 4*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 2*A(x)*x^n)^(n+1) = 0.
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 7*x^3 + 25*x^4 + 85*x^5 + 301*x^6 + 1086*x^7 + 3927*x^8 + 14328*x^9 + 52724*x^10 + 194915*x^11 + 723845*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 2*A(x))^n = 1 - 4*x + 4*x^4 - 4*x^9 + 4*x^16 - 4*x^25 + 4*x^36 - 4*x^49 +- ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04761601613534030259384050896565071457116692089742172541...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 2*A)^n = 2*(Pi/2)^(1/4)/gamma(3/4) - 1 = 0.82715827631223364281448...
(V.2) Let A = A(exp(-2*Pi)) = 0.00187446330928756547025110339586987296984387228299321603...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 2*A)^n = 2*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 1 = 0.99253022912181427157991...
(V.3) Let A = A(-exp(-Pi)) = -0.03996785964385216049635981950386915887875531406265280233...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 2*A)^n = 2*Pi^(1/4)/gamma(3/4) - 1 = 1.1728696224266160291506...
(V.4) Let A = A(-exp(-2*Pi)) = -0.00186051333175936112600864666861119312780357024086759004...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 2*A)^n = 2*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 1 = 1.007469770975478182...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=0, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, (-1)^m * (x^m + 2*Ser(A))^m ) - 1 - 4*sum(m=1, #A, (-1)^m * x^(m^2) ), #A-1)/2 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Feb 09 2024
STATUS
approved