A370030 Table in which the g.f. of row n, R(n,x), satisfies Sum_{k=-oo..+oo} (x^k - n*R(n,x))^k = 1 - (n-2)*Sum_{k>=1} x^(k^2), for n >= 1, as read by antidiagonals.

1, 1, 1, 1, 2, 0, 1, 3, 3, -1, 1, 4, 8, 3, 2, 1, 5, 15, 19, 5, 15, 1, 6, 24, 53, 46, 39, 27, 1, 7, 35, 111, 185, 161, 206, -1, 1, 8, 48, 199, 506, 711, 799, 697, -76, 1, 9, 63, 323, 1117, 2379, 3270, 4021, 1656, 19, 1, 10, 80, 489, 2150, 6335, 12083, 17297, 17932, 3208, 719, 1, 11, 99, 703, 3761, 14349, 37222, 67531, 95108, 71311, 8727, 1687

Offset: 1

Author

Paul D. Hanna, Feb 10 2024

Keywords

• sign
• tabl

Comments

A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

Examples

			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).
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..1275
• Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Crossrefs

Cf. A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.

Cf. A370041, A370020 (dual table).

Programs

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(""))

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.