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A370039 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 9*A(x))^n = 1 - 7*Sum_{n>=1} x^(n^2).

%I A370039 #13 Feb 16 2025 08:34:06
%S A370039 1,9,80,703,6130,53351,466315,4118167,36941188,337853203,3155619199,
%T A370039 30087573015,292226014968,2882482639376,28783571541579,
%U A370039 290149337803965,2945978857054165,30080058358496842,308542728377796463,3177317808394936571,32835881264222087409,340467815173685043729
%N A370039 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 9*A(x))^n = 1 - 7*Sum_{n>=1} x^(n^2).
%C A370039 A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%H A370039 Paul D. Hanna, <a href="/A370039/b370039.txt">Table of n, a(n) for n = 1..326</a>
%H A370039 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370039 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370039 (1) Sum_{n=-oo..+oo} (x^n - 9*A(x))^n = 1 - 7*Sum_{n>=1} x^(n^2).
%F A370039 (2) Sum_{n=-oo..+oo} x^n * (x^n + 9*A(x))^(n-1) = 1 - 7*Sum_{n>=1} x^(n^2).
%F A370039 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 9*A(x))^n = 0.
%F A370039 (4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 9*x^n*A(x))^n = 1 - 7*Sum_{n>=1} x^(n^2).
%F A370039 (5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 9*x^n*A(x))^(n+1) = 1 - 7*Sum_{n>=1} x^(n^2).
%F A370039 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 9*x^n*A(x))^n = 0.
%e A370039 G.f.: A(x) = x + 9*x^2 + 80*x^3 + 703*x^4 + 6130*x^5 + 53351*x^6 + 466315*x^7 + 4118167*x^8 + 36941188*x^9 + 337853203*x^10 + 3155619199*x^11 + ...
%e A370039 where
%e A370039 Sum_{n=-oo..+oo} (x^n - 8*A(x))^n = 1 - 7*x - 7*x^4 - 7*x^9 - 7*x^16 - 7*x^25 - 7*x^36 - 7*x^49 - ...
%e A370039 SPECIAL VALUES.
%e A370039 (V.1) Let A = A(exp(-Pi)) = 0.07041342765468695859173243504212855904085321490660808668...
%e A370039 then Sum_{n=-oo..+oo} (exp(-n*Pi) - 9*A)^n = (9 - 7*Pi^(1/4)/gamma(3/4))/2 = 0.69747816075342194898639...
%e A370039 (V.2) Let A = A(exp(-2*Pi)) = 0.001899358496977867055016493259704554658290299283307899768...
%e A370039 then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 9*A)^n = (9 - 7*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.98692790079291318133312...
%e A370039 (V.3) Let A = A(-exp(-Pi)) = -0.03108273985731889208644710399967055047528520340415555251...
%e A370039 then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 9*A)^n = (9 - 7*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.302473016453591125074...
%e A370039 (V.4) Let A = A(-exp(-2*Pi)) = -0.001836569230890760040434767580223720991124539653197115902...
%e A370039 then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 9*A)^n = (9 - 7*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.013072099036825024735...
%o A370039 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370039 A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 9*Ser(A))^m ) - 1 + 7*sum(m=1,#A, x^(m^2) ), #A-1)/9 ); A[n+1]}
%o A370039 for(n=1,30, print1(a(n),", "))
%Y A370039 Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370043.
%K A370039 nonn
%O A370039 1,2
%A A370039 _Paul D. Hanna_, Feb 10 2024