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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.

%I A370030 #8 Feb 16 2025 08:34:06
%S A370030 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,
%T A370030 35,111,185,161,206,-1,1,8,48,199,506,711,799,697,-76,1,9,63,323,1117,
%U A370030 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
%N 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.
%C A370030 A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%H A370030 Paul D. Hanna, <a href="/A370030/b370030.txt">Table of n, a(n) for n = 1..1275</a>
%H A370030 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370030 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370030 (1) Sum_{k=-oo..+oo} (x^k - n*R(n,x))^k = 1 - (n-2)*Sum_{k>=1} x^(k^2).
%F A370030 (2) Sum_{k=-oo..+oo} x^k * (x^k + n*R(n,x))^(k-1) = 1 - (n-2)*Sum_{k>=1} x^(k^2).
%F A370030 (3) Sum_{k=-oo..+oo} (-1)^k * x^k * (x^k - n*R(n,x))^k = 0.
%F A370030 (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).
%F A370030 (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).
%F A370030 (6) Sum_{k=-oo..+oo} (-1)^k * x^(k*(k-1)) / (1 - n*R(n,x)*x^k)^k = 0.
%e A370030 This table of coefficients T(n,k) of x^k in R(n,x), n >= 1, k >= 1, begins:
%e A370030 A370031: [1,  1,  0,  -1,    2,    15,     27,      -1,      -76, ...];
%e A370030 A355868: [1,  2,  3,   3,    5,    39,    206,     697,     1656, ...];
%e A370030 A370033: [1,  3,  8,  19,   46,   161,    799,    4021,    17932, ...];
%e A370030 A370034: [1,  4, 15,  53,  185,   711,   3270,   17297,    95108, ...];
%e A370030 A370035: [1,  5, 24, 111,  506,  2379,  12083,   67531,   406284, ...];
%e A370030 A370036: [1,  6, 35, 199, 1117,  6335,  37222,  230809,  1515784, ...];
%e A370030 A370037: [1,  7, 48, 323, 2150, 14349,  97431,  681857,  4956116, ...];
%e A370030 A370038: [1,  8, 63, 489, 3761, 28911, 224174, 1768801, 14298852, ...];
%e A370030 A370039: [1,  9, 80, 703, 6130, 53351, 466315, 4118167, 36941188, ...];
%e A370030 A370043: [1, 10, 99, 971, 9461, 91959, 895518, 8775161, 86870264, ...]; ...
%e A370030 ...
%e A370030 where the n-th row function R(n,x) satisfies
%e A370030 Sum_{k=-oo..+oo} (x^k - n*R(n,x))^k = 1 - (n-2)*Sum_{k>=1} x^(k^2).
%o A370030 (PARI) {T(n,k) = my(A=[0,1]); for(i=0,k, A = concat(A,0);
%o A370030 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]}
%o A370030 for(n=1,12, for(k=1,10, print1(T(n,k),", "));print(""))
%Y A370030 Cf. A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.
%Y A370030 Cf. A370041, A370020 (dual table).
%K A370030 sign,tabl
%O A370030 1,5
%A A370030 _Paul D. Hanna_, Feb 10 2024