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SPECIAL VALUES.
(V.1) Let A = A(exp(-Pi)) = 0.05266628449954901094912490050067062239110765179054552678...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-n*Pi) + 4*A)^n = 3*(Pi/2)^(1/4)/gamma(3/4) - 2 = 0.74073741446835046422172...
(V.2) Let A = A(exp(-2*Pi)) = 0.001881517053093894919707587041659521876650213322334450878...
then Sum_{n=-oo..+oo} (-1)^n * (exp(-2*n*Pi) + 4*A)^n = 3*2^(1/8)*(Pi/2)^(1/4)/gamma(3/4) - 2 = 0.988795343682721407369...
(V.3) Let A = A(-exp(-Pi)) = -0.03699687105031477666227946508842289849689211763245984347...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-n*Pi) + 4*A)^n = 3*Pi^(1/4)/gamma(3/4) - 2 = 1.259304433639924043725...
(V.4) Let A = A(-exp(-2*Pi)) = -0.0018536159060139689998658447922411419770684746756327600438...
then Sum_{n=-oo..+oo} (-1)^n * ((-1)^n*exp(-2*n*Pi) + 4*A)^n = 3*sqrt(2 + sqrt(2))/2 * Pi^(1/4)/gamma(3/4) - 2 = 1.011204656463217273143...
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Paul D. Hanna, <a href="/A370024/b370024_1.txt">Table of n, a(n) for n = 1..401</a>
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Paul D. Hanna, <a href="/A370024/b370024_1.txt">Table of n, a(n) for n = 1..401</a>
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Expansion of g.f. A(x) satisfying Sum_{n=-oo..+oo} (-1)^n * (x^n + 4*A(x))^n = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
1, 4, 19, 99, 529, 2853, 15566, 85879, 477716, 2674070, 15047671, 85063429, 482733230, 2748703604, 15697194139, 89875431754, 515774659357, 2966016776556, 17088046518051, 98614323921685, 569967829487533, 3298876334401503, 19117753534875276, 110922240116613681, 644276475406441599
1,2
A related function is theta_4(x) = 1 + 2*Sum_{n>=1} (-1)^n * x^(n^2).
Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/JacobiThetaFunctions.html">Jacobi Theta Functions</a>
G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.
(1) Sum_{n=-oo..+oo} (-1)^n * (x^n + 4*A(x))^n = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(2) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 4*A(x))^(n-1) = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(3) Sum_{n=-oo..+oo} (-1)^n * x^n * (x^n + 4*A(x))^n = 0.
(4) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^n)^n = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(5) Sum_{n=-oo..+oo} (-1)^n * x^(n^2) / (1 + 4*A(x)*x^n)^(n+1) = 1 + 6*Sum_{n>=1} (-1)^n * x^(n^2).
(6) Sum_{n=-oo..+oo} (-1)^n * x^(n*(n+1)) / (1 + 4*A(x)*x^n)^(n+1) = 0.
G.f.: A(x) = x + 4*x^2 + 19*x^3 + 99*x^4 + 529*x^5 + 2853*x^6 + 15566*x^7 + 85879*x^8 + 477716*x^9 + 2674070*x^10 + 15047671*x^11 + 85063429*x^12 + ...
where
Sum_{n=-oo..+oo} (-1)^n * (x^n + 4*A(x))^n = 1 - 6*x + 6*x^4 - 6*x^9 + 6*x^16 - 6*x^25 + 6*x^36 - 6*x^49 +- ...
(PARI) {a(n) = my(A=[0, 1]); for(i=1, n, A = concat(A, 0);
A[#A] = polcoeff( sum(m=-#A, #A, (-1)^m * (x^m + 4*Ser(A))^m ) - 1 - 6*sum(m=1, #A, (-1)^m * x^(m^2) ), #A-1)/4 ); A[n+1]}
for(n=1, 30, print1(a(n), ", "))
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Paul D. Hanna, Feb 09 2024
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