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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.
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%I #7 Feb 10 2024 12:35:09

%S 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 35,111,185,161,206,-1,1,8,48,199,506,711,799,697,-76,1,9,63,323,1117,

%U 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 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 A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).

%H Paul D. Hanna, <a href="/A370030/b370030.txt">Table of n, a(n) for n = 1..1275</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) Sum_{k=-oo..+oo} (x^k - n*R(n,x))^k = 1 - (n-2)*Sum_{k>=1} x^(k^2).

%F (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 (3) Sum_{k=-oo..+oo} (-1)^k * x^k * (x^k - n*R(n,x))^k = 0.

%F (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 (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 (6) Sum_{k=-oo..+oo} (-1)^k * x^(k*(k-1)) / (1 - n*R(n,x)*x^k)^k = 0.

%e This table of coefficients T(n,k) of x^k in R(n,x), n >= 1, k >= 1, begins:

%e A370031: [1, 1, 0, -1, 2, 15, 27, -1, -76, ...];

%e A355868: [1, 2, 3, 3, 5, 39, 206, 697, 1656, ...];

%e A370033: [1, 3, 8, 19, 46, 161, 799, 4021, 17932, ...];

%e A370034: [1, 4, 15, 53, 185, 711, 3270, 17297, 95108, ...];

%e A370035: [1, 5, 24, 111, 506, 2379, 12083, 67531, 406284, ...];

%e A370036: [1, 6, 35, 199, 1117, 6335, 37222, 230809, 1515784, ...];

%e A370037: [1, 7, 48, 323, 2150, 14349, 97431, 681857, 4956116, ...];

%e A370038: [1, 8, 63, 489, 3761, 28911, 224174, 1768801, 14298852, ...];

%e A370039: [1, 9, 80, 703, 6130, 53351, 466315, 4118167, 36941188, ...];

%e A370043: [1, 10, 99, 971, 9461, 91959, 895518, 8775161, 86870264, ...]; ...

%e ...

%e where the n-th row function R(n,x) satisfies

%e Sum_{k=-oo..+oo} (x^k - n*R(n,x))^k = 1 - (n-2)*Sum_{k>=1} x^(k^2).

%o (PARI) {T(n,k) = my(A=[0,1]); for(i=0,k, A = concat(A,0);

%o 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 for(n=1,12, for(k=1,10, print1(T(n,k),", "));print(""))

%Y Cf. A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.

%Y Cf. A370041, A370020 (dual table).

%K sign,tabl

%O 1,5

%A _Paul D. Hanna_, Feb 10 2024