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A370031 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - A(x))^n = Sum_{n>=0} x^(n^2).

%I A370031 #15 Feb 16 2025 08:34:06
%S A370031 1,1,0,-1,2,15,27,-1,-76,19,719,1687,184,-5976,-3749,44093,130933,
%T A370031 42026,-512833,-667101,2976177,11391169,6608432,-45604863,-87819235,
%U A370031 202544340,1053407806,922859161,-4085924365,-10600384406,12656739909,100646660458,121472828448,-360976456530
%N A370031 Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (x^n - A(x))^n = Sum_{n>=0} x^(n^2).
%C A370031 A related function is theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).
%H A370031 Paul D. Hanna, <a href="/A370031/b370031.txt">Table of n, a(n) for n = 1..531</a>
%H A370031 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
%F A370031 G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
%F A370031 (1) Sum_{n=-oo..+oo} (x^n - A(x))^n = Sum_{n>=0} x^(n^2).
%F A370031 (2) Sum_{n=-oo..+oo} x^n * (x^n + A(x))^(n-1) = Sum_{n>=0} x^(n^2).
%F A370031 (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n - A(x))^n = 0.
%F A370031 (4) Sum_{n=-oo..+oo} x^(n^2) / (1 - x^n*A(x))^n = Sum_{n>=0} x^(n^2).
%F A370031 (5) Sum_{n=-oo..+oo} x^(n^2) / (1 + x^n*A(x))^(n+1) = Sum_{n>=0} x^(n^2).
%F A370031 (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 - x^n*A(x))^n = 0.
%e A370031 G.f.: A(x) = 1 + x - x^3 + 2*x^4 + 15*x^5 + 27*x^6 - x^7 - 76*x^8 + 19*x^9 + 719*x^10 + 1687*x^11 + 184*x^12 - 5976*x^13 - 3749*x^14 + 44093*x^15 + ...
%e A370031 where
%e A370031 Sum_{n=-oo..+oo} (x^n - A(x))^n = 1 + x + x^4 + x^9 + x^16 + x^25 + x^36 + x^49 + ...
%e A370031 SPECIAL VALUES.
%e A370031 (V.1) Let A = A(exp(-Pi)) = 0.04507828029039130528308497098432879536368681539259286273...
%e A370031 then Sum_{n=-oo..+oo} (exp(-n*Pi) - A)^n = (1 + Pi^(1/4)/gamma(3/4))/2 = 1.0432174056066540...
%e A370031 (V.2) Let A = A(exp(-2*Pi)) = 0.001870930061948701432816606547007172908053584772650237678...
%e A370031 then Sum_{n=-oo..+oo} (exp(-2*n*Pi) - A)^n = (1 + sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4))/2 = 1.00186744274386954...
%e A370031 (V.3) Let A = A(-exp(-Pi)) = -0.04135017416264159536574596265267969182735801577042441264...
%e A370031 then Sum_{n=-oo..+oo} ((-1)^n*exp(-n*Pi) - A)^n = (1 + (Pi/2)^(1/4)/gamma(3/4))/2 = 0.9567895690780584107...
%e A370031 (V.4) Let A = A(-exp(-2*Pi)) = -0.001863955401558124515592555303127910405358304631205735085...
%e A370031 then Sum_{n=-oo..+oo} ((-1)^n*exp(-2*n*Pi) - A)^n = (1 + 2^(1/8)*(Pi/2)^(1/4)/gamma(3/4))/2 = 0.99813255728045356...
%o A370031 (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);
%o A370031 A[#A] = polcoeff( sum(n=-#A,#A, (x^n - Ser(A))^n ) - sum(n=0,#A, x^(n^2) ), #A-1) ); A[n+1]}
%o A370031 for(n=1,40, print1(a(n),", "))
%Y A370031 Cf. A370041, A370030, A355868, A370033, A370034, A370035, A370036, A370037, A370038, A370039, A370043.
%K A370031 sign
%O A370031 1,5
%A A370031 _Paul D. Hanna_, Feb 10 2024