OFFSET
1,5
COMMENTS
A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..531
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} (x^n - A(x))^n = Sum_{n>=0} x^(n^2).
(2) Sum_{n=-oo..+oo} x^n * (x^n + A(x))^(n-1) = Sum_{n>=0} x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - A(x))^n = 0.
(4) Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n*A(x))^n = Sum_{n>=0} x^(n^2).
(5) Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n*A(x))^(n+1) = Sum_{n>=0} x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - x^n*A(x))^n = 0.
EXAMPLE
G.f.: A(x) = 1 + x - x^3 + 2*x^4 + 15*x^5 + 27*x^6 - x^7 - 76*x^8 + 19*x^9 + 719*x^10 + 1687*x^11 + 184*x^12 - 5976*x^13 - 3749*x^14 + 44093*x^15 + ...
where
Sum_{n=-oo..+oo} (x^n - A(x))^n = 1 + x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + ...
SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.04507828029039130528308497098432879536368681539259286273...
then Sum_{n=-oo..+oo} (exp(-n*Pi) - A)^n = (1 + Pi^(1/4)/gamma(3/4))/2 = 1.0432174056066540...
(V.2) Let A = A(exp(-2*Pi)) = 0.001870930061948701432816606547007172908053584772650237678...
then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - A)^n = (1 + sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 1.00186744274386954...
(V.3) Let A = A(-exp(-Pi)) = -0.04135017416264159536574596265267969182735801577042441264...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - A)^n = (1 + (Pi/2)^(1/4)/gamma(3/4))/2 = 0.9567895690780584107...
(V.4) Let A = A(-exp(-2*Pi)) = -0.001863955401558124515592555303127910405358304631205735085...
then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - A)^n = (1 + 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 0.99813255728045356...
PROG
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(n=-#A, #A, (x^n - Ser(A))^n ) - sum(n=0, #A, x^(n^2) ), #A-1) ); A[n+1]}
for(n=1, 40, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Feb 10 2024
STATUS
approved