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A370043 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 10*A(x))^n = 1 - 8*Sum_{n>=1} x^(n^2).

%I A370043 #14 Feb 16 2025 08:34:06
%S A370043 1,10,99,971,9461,91959,895518,8775161,86870264,871650208,8884142855,
%T A370043 92061370003,969550433086,10363557226896,112215017274331,
%U A370043 1228207449471086,13561137797537413,150791851996365182,1686274213530482843,18945675318778308411,213704510012147008821
%N A370043 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 10*A(x))^n = 1 - 8*Sum_{n>=1} x^(n^2).
%C A370043 A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%H A370043 Paul D. Hanna, <a href="/A370043/b370043.txt">Table of n, a(n) for n = 1..326</a>
%H A370043 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370043 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370043 (1) Sum_{n=-oo..+oo} (x^n - 10*A(x))^n = 1 - 8*Sum_{n>=1} x^(n^2).
%F A370043 (2) Sum_{n=-oo..+oo} x^n * (x^n + 10*A(x))^(n-1) = 1 - 8*Sum_{n>=1} x^(n^2).
%F A370043 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 10*A(x))^n = 0.
%F A370043 (4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 10*x^n*A(x))^n = 1 - 8*Sum_{n>=1} x^(n^2).
%F A370043 (5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 10*x^n*A(x))^(n+1) = 1 - 8*Sum_{n>=1} x^(n^2).
%F A370043 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 10*x^n*A(x))^n = 0.
%e A370043 G.f.: A(x) = x + 10*x^2 + 99*x^3 + 971*x^4 + 9461*x^5 + 91959*x^6 + 895518*x^7 + 8775161*x^8 + 86870264*x^9 + 871650208*x^10 + 8884142855*x^11 + ...
%e A370043 where
%e A370043 Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 8*x - 8*x^4 - 8*x^9 - 8*x^16 - 8*x^25 - 8*x^36 - 8*x^49 - ...
%e A370043 SPECIAL VALUES.
%e A370043 (V.1) Let A = A(exp(-Pi)) = 0.07572861892245027340976642864415638836692678958408803376...
%e A370043 then Sum_{n=-oo..+oo} (exp(-n*Pi) - 10*A)^n = 5 - 4*Pi^(1/4)/gamma(3/4) = 0.6542607551467679416987...
%e A370043 (V.2) Let A = A(exp(-2*Pi)) = 0.001902972911784356118532074933211699956337964100195554269...
%e A370043 then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 10*A)^n = 5 - 4*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) = 0.9850604580490436358...
%e A370043 (V.3) Let A = A(-exp(-Pi)) = -0.03014664142938059660934561948726688645121488051083843222...
%e A370043 then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 10*A)^n = 5 - 4*(Pi/2)^(1/4)/gamma(3/4) = 1.34568344737553271437...
%e A370043 (V.4) Let A = A(-exp(-2*Pi)) = -0.001833202439114209450155973975718938793478260093149995057...
%e A370043 then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 10*A)^n = 5 - 4*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) = 1.0149395417563714568...
%o A370043 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370043 A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 10*Ser(A))^m ) - 1 + 8*sum(m=1,#A, x^(m^2) ), #A-1)/10 ); A[n+1]}
%o A370043 for(n=1,30, print1(a(n),", "))
%Y A370043 Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039.
%K A370043 nonn
%O A370043 1,2
%A A370043 _Paul D. Hanna_, Feb 10 2024