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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.
13
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
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
nonn,tabl,changed
AUTHOR
Paul D. Hanna, Feb 09 2024
STATUS
approved