A370020 Table in which the g.f. of row n, 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), for n >= 1, as read by antidiagonals.

1, 1, 1, 1, 2, 4, 1, 3, 7, 9, 1, 4, 12, 25, 22, 1, 5, 19, 53, 85, 63, 1, 6, 28, 99, 234, 301, 155, 1, 7, 39, 169, 529, 1041, 1086, 415, 1, 8, 52, 269, 1054, 2853, 4711, 3927, 1124, 1, 9, 67, 405, 1917, 6667, 15566, 21573, 14328, 2957, 1, 10, 84, 583, 3250, 13893, 42627, 85879, 99484, 52724, 8047, 1, 11, 103, 809, 5209, 26541, 101830, 275211, 477716, 461657, 194915, 21817

Offset: 1

Author

Paul D. Hanna, Feb 09 2024

Keywords

• nonn
• tabl

Comments

A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

Examples

			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).
		

Links

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

Crossrefs

Cf. A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370029, A370042.

Cf. A370040, A370030 (dual table).

Programs

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(""))

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.