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A370038 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 6*Sum_{n>=1} x^(n^2).

%I A370038 #13 Feb 16 2025 08:34:06
%S A370038 1,8,63,489,3761,28911,224174,1768801,14298852,118834966,1014912939,
%T A370038 8876489811,79106007766,714758437500,6521121292423,59905861779190,
%U A370038 553172777516749,5129986605394544,47761053650028335,446350549038171483,4186889953961917077,39416115485839527945
%N A370038 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 6*Sum_{n>=1} x^(n^2).
%C A370038 A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%H A370038 Paul D. Hanna, <a href="/A370038/b370038.txt">Table of n, a(n) for n = 1..326</a>
%H A370038 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370038 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370038 (1) Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 6*Sum_{n>=1} x^(n^2).
%F A370038 (2) Sum_{n=-oo..+oo} x^n * (x^n + 8*A(x))^(n-1) = 1 - 6*Sum_{n>=1} x^(n^2).
%F A370038 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 8*A(x))^n = 0.
%F A370038 (4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 8*x^n*A(x))^n = 1 - 6*Sum_{n>=1} x^(n^2).
%F A370038 (5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 8*x^n*A(x))^(n+1) = 1 - 6*Sum_{n>=1} x^(n^2).
%F A370038 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 8*x^n*A(x))^n = 0.
%e A370038 G.f.: A(x) = x + 8*x^2 + 63*x^3 + 489*x^4 + 3761*x^5 + 28911*x^6 + 224174*x^7 + 1768801*x^8 + 14298852*x^9 + 118834966*x^10 + 1014912939*x^11 + ...
%e A370038 where
%e A370038 Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 6*x - 6*x^4 - 6*x^9 - 6*x^16 - 6*x^25 - 6*x^36 - 6*x^49 - ...
%e A370038 SPECIAL VALUES.
%e A370038 (V.1) Let A = A(exp(-Pi)) = 0.06579433445460281447496748523290398966344297589844019028...
%e A370038 then Sum_{n=-oo..+oo} (exp(-n*Pi) - 8*A)^n = 4 - 3*Pi^(1/4)/gamma(3/4) = 0.740695566360075956...
%e A370038 (V.2) Let A = A(exp(-2*Pi)) = 0.001895757786183755555448115532175643265455444051246465664...
%e A370038 then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 8*A)^n = 4 - 3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.98879534353678272...
%e A370038 (V.3) Let A = A(-exp(-Pi)) = -0.03207876150064786089070312769117792591667175850120792604...
%e A370038 then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 8*A)^n = 4 - 3*(Pi/2)^(1/4)/gamma(3/4) = 1.25926258553164953...
%e A370038 (V.4) Let A = A(-exp(-2*Pi)) = -0.001839948412029108042031275075360099309960919616491079407...
%e A370038 then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 8*A)^n = 4 - 3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.01120465631727859...
%o A370038 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370038 A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 8*Ser(A))^m ) - 1 + 6*sum(m=1,#A, x^(m^2) ), #A-1)/8 ); A[n+1]}
%o A370038 for(n=1,30, print1(a(n),", "))
%Y A370038 Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370039, A370043.
%K A370038 nonn
%O A370038 1,2
%A A370038 _Paul D. Hanna_, Feb 10 2024