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A370035 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 5*A(x))^n = 1 - 3*Sum_{n>=1} x^(n^2).

%I A370035 #13 Feb 16 2025 08:34:06
%S A370035 1,5,24,111,506,2379,12083,67531,406284,2531203,15866775,98883303,
%T A370035 612775096,3798083196,23698615411,149450139421,953022469813,
%U A370035 6132672546362,39706366904663,258032916789711,1680779512045521,10970344718827785,71764325720800072,470691291168007709
%N A370035 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 5*A(x))^n = 1 - 3*Sum_{n>=1} x^(n^2).
%C A370035 A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%H A370035 Paul D. Hanna, <a href="/A370035/b370035.txt">Table of n, a(n) for n = 1..401</a>
%H A370035 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370035 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370035 (1) Sum_{n=-oo..+oo} (x^n - 5*A(x))^n = 1 - 3*Sum_{n>=1} x^(n^2).
%F A370035 (2) Sum_{n=-oo..+oo} x^n * (x^n + 5*A(x))^(n-1) = 1 - 3*Sum_{n>=1} x^(n^2).
%F A370035 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 5*A(x))^n = 0.
%F A370035 (4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 5*x^n*A(x))^n = 1 - 3*Sum_{n>=1} x^(n^2).
%F A370035 (5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 5*x^n*A(x))^(n+1) = 1 - 3*Sum_{n>=1} x^(n^2).
%F A370035 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 5*x^n*A(x))^n = 0.
%e A370035 G.f.: A(x) = x + 5*x^2 + 24*x^3 + 111*x^4 + 506*x^5 + 2379*x^6 + 12083*x^7 + 67531*x^8 + 406284*x^9 + 2531203*x^10 + 15866775*x^11 + 98883303*x^12 + ...
%e A370035 where
%e A370035 Sum_{n=-oo..+oo} (x^n - 5*A(x))^n = 1 - 3*x - 3*x^4 - 3*x^9 - 3*x^16 - 3*x^25 - 3*x^36 - 3*x^49 - ...
%e A370035 \SPECIAL VALUES.
%e A370035 (V.1) Let A = A(exp(-Pi)) = 0.05497127752043377386868704294930896868077597772598908285...
%e A370035 then Sum_{n=-oo..+oo} (exp(-n*Pi) - 5*A)^n = (5 - 3*Pi^(1/4)/gamma(3/4) )/2 = 0.870347783180037978...
%e A370035 (V.2) Let A = A(exp(-2*Pi)) = 0.001885037102906729934432374294398706956703235597857256076...
%e A370035 then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 5*A)^n = (5 - 3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.994397671768391...
%e A370035 (V.3) Let A = A(-exp(-Pi)) = -0.03548990756971248576955893224372969073755967165800772531...
%e A370035 then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 5*A)^n = (5 - 3*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.12963129276582476...
%e A370035 (V.4) Let A = A(-exp(-2*Pi)) = -0.001850160979277236538611428135062916090397865766804127684...
%e A370035 then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 5*A)^n = (5 - 3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.005602328158639...
%o A370035 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370035 A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 5*Ser(A))^m ) - 1 + 3*sum(m=1,#A, x^(m^2) ), #A-1)/5 ); A[n+1]}
%o A370035 for(n=1,30, print1(a(n),", "))
%Y A370035 Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370036, A370037, A370038, A370039, A370043.
%K A370035 nonn
%O A370035 1,2
%A A370035 _Paul D. Hanna_, Feb 10 2024