%I #9 Feb 12 2024 19:31:25

%S 1,5,28,169,1054,6667,42627,275211,1791132,11731613,77242391,

%T 510826889,3391115560,22586150402,150866419771,1010290295683,

%U 6780795305121,45602955247738,307252705965207,2073546683753911,14014659243408481,94851805738129599,642767262413178788,4360774590348465669

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

%C A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).

%H Paul D. Hanna, <a href="/A370025/b370025.txt">Table of n, a(n) for n = 1..401</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>

%F G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

%F (1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 5*A(x))^n = 1 + 7*Sum_{n>=1} (-1)^n * x^(n^2).

%F (2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 5*A(x))^(n-1) = 1 + 7*Sum_{n>=1} (-1)^n * x^(n^2).

%F (3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 5*A(x))^n = 0.

%F (4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 5*A(x)*x^n)^n = 1 + 7*Sum_{n>=1} (-1)^n * x^(n^2).

%F (5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 5*A(x)*x^n)^(n+1) = 1 + 7*Sum_{n>=1} (-1)^n * x^(n^2).

%F (6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 5*A(x)*x^n)^(n+1) = 0.

%e G.f.: A(x) = x + 5*x^2 + 28*x^3 + 169*x^4 + 1054*x^5 + 6667*x^6 + 42627*x^7 + 275211*x^8 + 1791132*x^9 + 11731613*x^10 + 77242391*x^11 + 510826889*x^12 + ...

%e where

%e Sum_{n=-oo..+oo} (-1)^n * (x^n + 5*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 SPECIAL VALUES.

%e (V.1) Let A = A(exp(-Pi)) = 0.05561899448885311185126383683351896617798185829954077412...

%e then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 5*A)^n = (7*(Pi/2)^(1/4)/gamma(3/4) - 5)/2 = 0.69752698354640887492534...

%e (V.2) Let A = A(exp(-2*Pi)) = 0.001885063870555508038278982205994616246272805466135524875...

%e then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 5*A)^n = (7*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 5)/2 = 0.986927900963174975264...

%e (V.3) Let A = A(-exp(-Pi)) = -0.03567173485605183837843763169616623725553901880108539739...

%e then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 5*A)^n = (7*Pi^(1/4)/gamma(3/4) - 5)/2 = 1.302521839246578051013...

%e (V.4) Let A = A(-exp(-2*Pi)) = -0.0018536159060139689998658447922411419770684746756327600438...

%e then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 5*A)^n = (7*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 5)/2 = 1.01307209920708681866...

%o (PARI) {a(n) = my(A=[0,1]); for(i=1,n, A = concat(A,0);

%o A[#A] = polcoeff( sum(m=-#A,#A, (-1)^m * (x^m + 5*Ser(A))^m ) - 1 - 7*sum(m=1,#A, (-1)^m * x^(m^2) ), #A-1)/5 ); A[n+1]}

%o for(n=1,30, print1(a(n),", "))

%Y Cf. A370020, A370021, A370022, A370023, A370024, A370026, A370027, A370028, A370029, A370042.

%K nonn

%O 1,2

%A _Paul D. Hanna_, Feb 09 2024