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..1275
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_{k=-oo..+oo} (x^k - n*R(n,x))^k = 1 - (n-2)*Sum_{k>=1} x^(k^2).
(2) Sum_{k=-oo..+oo} x^k * (x^k + n*R(n,x))^(k-1) = 1 - (n-2)*Sum_{k>=1} x^(k^2).
(3) Sum_{k=-oo..+oo} (-1)^k * x^k * (x^k - n*R(n,x))^k = 0.
(4) Sum_{k=-oo..+oo} x^(k^2) / (1 - n*R(n,x)*x^k)^k = 1 - (n-2)*Sum_{k>=1} x^(k^2).
(5) Sum_{k=-oo..+oo} x^(k^2) / (1 + n*R(n,x)*x^k)^(k+1) = 1 - (n-2)*Sum_{k>=1} x^(k^2).
(6) Sum_{k=-oo..+oo} (-1)^k * x^(k*(k-1)) / (1 - n*R(n,x)*x^k)^k = 0.
EXAMPLE
This table of coefficients T(n,k) of x^k in R(n,x), n >= 1, k >= 1, begins:
A370031: [1, 1, 0, -1, 2, 15, 27, -1, -76, ...];
A355868: [1, 2, 3, 3, 5, 39, 206, 697, 1656, ...];
A370033: [1, 3, 8, 19, 46, 161, 799, 4021, 17932, ...];
A370034: [1, 4, 15, 53, 185, 711, 3270, 17297, 95108, ...];
A370035: [1, 5, 24, 111, 506, 2379, 12083, 67531, 406284, ...];
A370036: [1, 6, 35, 199, 1117, 6335, 37222, 230809, 1515784, ...];
A370037: [1, 7, 48, 323, 2150, 14349, 97431, 681857, 4956116, ...];
A370038: [1, 8, 63, 489, 3761, 28911, 224174, 1768801, 14298852, ...];
A370039: [1, 9, 80, 703, 6130, 53351, 466315, 4118167, 36941188, ...];
A370043: [1, 10, 99, 971, 9461, 91959, 895518, 8775161, 86870264, ...]; ...
...
where the n-th row function R(n,x) satisfies
Sum_{k=-oo..+oo} (x^k - n*R(n,x))^k = 1 - (n-2)*Sum_{k>=1} x^(k^2).
PROG
(PARI) {T(n, k) = my(A=[0, 1]); for(i=0, k, A = concat(A, 0);
A[#A] = polcoeff( sum(m=-sqrtint(#A+1), #A, (x^m - n*Ser(A))^m ) - 1 + (n-2)*sum(m=1, sqrtint(#A+1), x^(m^2) ), #A-1)/n ); A[k+1]}
for(n=1, 12, for(k=1, 10, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Paul D. Hanna, Feb 10 2024
STATUS
approved