This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370020 #11 Feb 16 2025 08:34:06
%S A370020 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,
%T A370020 155,1,7,39,169,529,1041,1086,415,1,8,52,269,1054,2853,4711,3927,1124,
%U A370020 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
%N 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.
%C A370020 A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
%H A370020 Paul D. Hanna, <a href="/A370020/b370020.txt">Table of n, a(n) for n = 1..1275</a>
%H A370020 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370020 The n-th row g.f. R(n,x) = Sum_{k>=1} T(n,k)*x^k satisfies the following formulas.
%F A370020 (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).
%F A370020 (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).
%F A370020 (3) Sum_{k=-oo..+oo} (-1)^k * x^k * (x^k + n*R(n,x))^k = 0.
%F A370020 (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).
%F A370020 (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).
%F A370020 (6) Sum_{k=-oo..+oo} (-1)^k * x^(k*(k+1)) / (1 + n*R(n,x)*x^k)^(k+1) = 0.
%e A370020 This table of coefficients T(n,k) of x^k in R(n,x), n >= 1, k >= 1, begins:
%e A370020 A370021: [1, 1, 4, 9, 22, 63, 155, 415, ...];
%e A370020 A370022: [1, 2, 7, 25, 85, 301, 1086, 3927, ...];
%e A370020 A370023: [1, 3, 12, 53, 234, 1041, 4711, 21573, ...];
%e A370020 A370024: [1, 4, 19, 99, 529, 2853, 15566, 85879, ...];
%e A370020 A370025: [1, 5, 28, 169, 1054, 6667, 42627, 275211, ...];
%e A370020 A370026: [1, 6, 39, 269, 1917, 13893, 101830, 753255, ...];
%e A370020 A370027: [1, 7, 52, 405, 3250, 26541, 219311, 1828657, ...];
%e A370020 A370028: [1, 8, 67, 583, 5209, 47341, 435366, 4039863, ...];
%e A370020 A370029: [1, 9, 84, 809, 7974, 79863, 809131, 8270199, ...];
%e A370020 A370042: [1, 10, 103, 1089, 11749, 128637, 1423982, 15898231, ...];
%e A370020 ...
%e A370020 where the n-th row function R(n,x) satisfies
%e A370020 Sum_{k=-oo..+oo} (-1)^k * (x^k + n*R(n,x))^k = 1 + (n+2)*Sum_{k>=1} (-1)^k * x^(k^2).
%o A370020 (PARI) {T(n,k) = my(A=[0,1]); for(i=0,k, A = concat(A,0);
%o A370020 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]}
%o A370020 for(n=1,12, for(k=1,10, print1(T(n,k),", "));print(""))
%Y A370020 Cf. A370021, A370022, A370023, A370024, A370025, A370026, A370027, A370028, A370029, A370042.
%Y A370020 Cf. A370040, A370030 (dual table).
%K A370020 nonn,tabl
%O A370020 1,5
%A A370020 _Paul D. Hanna_, Feb 09 2024