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

1,5

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

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

Cf. A370040, A370030 (dual table).

Sequence in context: A163360 A348138 A076053 * A338530 A248212 A290824

Adjacent sequences: A370017 A370018 A370019 * A370021 A370022 A370023

Keyword

nonn,tabl

Author

Paul D. Hanna, Feb 09 2024

Status

approved