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A370033 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 3*A(x))^n = 1 - Sum_{n>=1} x^(n^2).

%I A370033 #13 Feb 16 2025 08:34:06
%S A370033 1,3,8,19,46,161,799,4021,17932,71311,268639,1045731,4464576,20500010,
%T A370033 95221503,429913365,1879365529,8112744634,35452835755,158833086233,
%U A370033 725458442577,3329609464605,15194309369384,68837584452055,311257278509193,1413730859134250,6469321177004978
%N A370033 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 3*A(x))^n = 1 - Sum_{n>=1} x^(n^2).
%C A370033 A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%H A370033 Paul D. Hanna, <a href="/A370033/b370033.txt">Table of n, a(n) for n = 1..451</a>
%H A370033 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370033 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370033 (1) Sum_{n=-oo..+oo} (x^n - 3*A(x))^n = 1 - Sum_{n>=1} x^(n^2).
%F A370033 (2) Sum_{n=-oo..+oo} x^n * (x^n + 3*A(x))^(n-1) = 1 - Sum_{n>=1} x^(n^2).
%F A370033 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 3*A(x))^n = 0.
%F A370033 (4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 3*x^n*A(x))^n = 1 - Sum_{n>=1} x^(n^2).
%F A370033 (5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 3*x^n*A(x))^(n+1) = 1 - Sum_{n>=1} x^(n^2).
%F A370033 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 3*x^n*A(x))^n = 0.
%e A370033 G.f.: A(x) = 1 + 3*x + 8*x^2 + 19*x^3 + 46*x^4 + 161*x^5 + 799*x^6 + 4021*x^7 + 17932*x^8 + 71311*x^9 + 268639*x^10 + 1045731*x^11 + 4464576*x^12 + ...
%e A370033 where
%e A370033 Sum_{n=-oo..+oo} (x^n - 3*A(x))^n = 1 - x - x^4 - x^9 - x^16 - x^25 - x^36 - x^49 - ...
%e A370033 SPECIAL VALUES.
%e A370033 (V.1) Let A = A(exp(-Pi)) = 0.04953636800560980886288845724196786482586224709976648461...
%e A370033 then Sum_{n=-oo..+oo} (exp(-n*Pi) - 3*A)^n = (3 - Pi^(1/4)/gamma(3/4))/2 = 0.956782594393345992...
%e A370033 (V.2) Let A = A(exp(-2*Pi)) = 0.001877957090194880545086201853719041435355287864597005509...
%e A370033 then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 3*A)^n = (3 - sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.998132557256130454...
%e A370033 (V.3) Let A = A(-exp(-Pi)) = -0.03819699447470815952471171970837842342724818247967540335...
%e A370033 then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 3*A)^n = (3 - (Pi/2)^(1/4)/gamma(3/4))/2 = 1.0432104309219415...
%e A370033 (V.4) Let A = A(-exp(-2*Pi)) = -0.001857032573904813918259314464039219802478066024973444789...
%e A370033 then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 3*A)^n = (3 - 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.001867442719546432...
%o A370033 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370033 A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 3*Ser(A))^m ) - 1 + sum(m=1,#A, x^(m^2) ), #A-1)/3 ); A[n+1]}
%o A370033 for(n=1,30, print1(a(n),", "))
%Y A370033 Cf. A370041, A370030, A370031, A355868, A370034, A370035, A370036, A370037, A370038, A370039, A370043.
%K A370033 nonn
%O A370033 1,2
%A A370033 _Paul D. Hanna_, Feb 10 2024