OFFSET
1,5
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..1275
Eric Weisstein's World of Mathematics, Jacobi Theta Functions
FORMULA
The n-th row g.f. R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfies the following formulas.
(1) Sum_{k=-oo..+oo} (-1)^k * (x^k + n*R(n,x))^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).
(2) Sum_{k=-oo..+oo} (-1)^k * x^k * (x^k + n*R(n,x))^(k-1) = 1 + (n+2)*Sum_{k>=1} (-1)^k * 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} (-1)^k * x^(k^2) / (1 + n*R(n,x)*x^k)^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).
(5) Sum_{k=-oo..+oo} (-1)^k * x^(k^2) / (1 + n*R(n,x)*x^k)^(k+1) = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).
(6) Sum_{k=-oo..+oo} (-1)^k * x^(k*(k+1)) / (1 + n*R(n,x)*x^k)^(k+1) = 0.
EXAMPLE
This table of coefficients T(n,k) of x^k in R(n,x), n >= 1, k >= 1, begins:
A370021: [1, 1, 4, 9, 22, 63, 155, 415, ...];
A370022: [1, 2, 7, 25, 85, 301, 1086, 3927, ...];
A370023: [1, 3, 12, 53, 234, 1041, 4711, 21573, ...];
A370024: [1, 4, 19, 99, 529, 2853, 15566, 85879, ...];
A370025: [1, 5, 28, 169, 1054, 6667, 42627, 275211, ...];
A370026: [1, 6, 39, 269, 1917, 13893, 101830, 753255, ...];
A370027: [1, 7, 52, 405, 3250, 26541, 219311, 1828657, ...];
A370028: [1, 8, 67, 583, 5209, 47341, 435366, 4039863, ...];
A370029: [1, 9, 84, 809, 7974, 79863, 809131, 8270199, ...];
A370042: [1, 10, 103, 1089, 11749, 128637, 1423982, 15898231, ...];
...
where the n-th row function R(n,x) satisfies
Sum_{k=-oo..+oo} (-1)^k * (x^k + n*R(n,x))^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * 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=-#A, #A, (-1)^m * (x^m + n*Ser(A))^m ) - 1 - (n+2)*sum(m=1, #A, (-1)^m * x^(m^2) ), #A-1)/n ); A[k+1]}
for(n=1, 12, for(k=1, 10, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
AUTHOR
Paul D. Hanna, Feb 09 2024
STATUS
approved