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A370037 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 7*A(x))^n = 1 - 5*Sum_{n>=1} x^(n^2).

%I A370037 #13 Feb 16 2025 08:34:06
%S A370037 1,7,48,323,2150,14349,97431,681857,4956116,37422943,291763607,
%T A370037 2327820547,18848767552,153975563934,1264733865463,10431060837749,
%U A370037 86368123241833,718121985169658,5997857713743011,50325664101701349,424138198629299217,3589151537280637957,30481898682409007792
%N A370037 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - 7*A(x))^n = 1 - 5*Sum_{n>=1} x^(n^2).
%C A370037 A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%H A370037 Paul D. Hanna, <a href="/A370037/b370037.txt">Table of n, a(n) for n = 1..326</a>
%H A370037 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370037 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370037 (1) Sum_{n=-oo..+oo} (x^n - 7*A(x))^n = 1 - 5*Sum_{n>=1} x^(n^2).
%F A370037 (2) Sum_{n=-oo..+oo} x^n * (x^n + 7*A(x))^(n-1) = 1 - 5*Sum_{n>=1} x^(n^2).
%F A370037 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - 7*A(x))^n = 0.
%F A370037 (4) Sum_{n=-oo..+oo} x^(n^2) / (1 - 7*x^n*A(x))^n = 1 - 5*Sum_{n>=1} x^(n^2).
%F A370037 (5) Sum_{n=-oo..+oo} x^(n^2) / (1 + 7*x^n*A(x))^(n+1) = 1 - 5*Sum_{n>=1} x^(n^2).
%F A370037 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - 7*x^n*A(x))^n = 0.
%e A370037 G.f.: A(x) = x + 7*x^2 + 48*x^3 + 323*x^4 + 2150*x^5 + 14349*x^6 + 97431*x^7 + 681857*x^8 + 4956116*x^9 + 37422943*x^10 + 291763607*x^11 + ...
%e A370037 where
%e A370037 Sum_{n=-oo..+oo} (x^n - 7*A(x))^n = 1 - 5*x - 5*x^4 - 5*x^9 - 5*x^16 - 5*x^25 - 5*x^36 - 5*x^49 - ...
%e A370037 SPECIAL VALUES.
%e A370037 (V.1) Let A = A(exp(-Pi)) = 0.06174306640715063509845961016774795661670689719654375131...
%e A370037 then Sum_{n=-oo..+oo} (exp(-n*Pi) - 7*A)^n = (7 - 5*Pi^(1/4)/gamma(3/4))/2 = 0.78391297196672996...
%e A370037 (V.2) Let A = A(exp(-2*Pi)) = 0.001892170701611855386420113452656768397809538392272023405...
%e A370037 then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - 7*A)^n = (7 - 5*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 0.99066278628065227...
%e A370037 (V.3) Let A = A(-exp(-Pi)) = -0.03314064170176376172583062314299135400117746157373562359...
%e A370037 then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - 7*A)^n = (7 - 5*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.2160521546097079...
%e A370037 (V.4) Let A = A(-exp(-2*Pi)) = -0.001843340051041967867985717685295689652563446679869985649...
%e A370037 then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - 7*A)^n = (7 - 5*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 1.00933721359773216...
%o A370037 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370037 A[#A] = polcoeff( sum(m=-#A,#A, (x^m - 7*Ser(A))^m ) - 1 + 5*sum(m=1,#A, x^(m^2) ), #A-1)/7 ); A[n+1]}
%o A370037 for(n=1,30, print1(a(n),", "))
%Y A370037 Cf. A370041, A370030, A370031, A355868, A370033, A370034, A370035, A370036, A370038, A370039, A370043.
%K A370037 nonn
%O A370037 1,2
%A A370037 _Paul D. Hanna_, Feb 10 2024