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%I #12 Dec 10 2022 14:33:35
%S 8,32,128,0,-9216,-94208,-671744,-3014656,1245184,171704320,
%T 1756364800,8338276352,-26013073408,-946201427968,-10033714692096,
%U -56471303749632,43465874341888,4967278927937536,61805829224923136,423546310109429760,713014908152709120,-24149207336980840448
%N a(n) = coefficient of x^(2*n) in S(x) defined by: C(x) + i*S(x) = Sum_{n=-oo..+oo} i^n * (2*x)^(n^2) * F(x)^n, where F(x) is the g.f. of A357787 such that C(x)^2 + S(x)^2 = 1.
%H Paul D. Hanna, <a href="/A357789/b357789.txt">Table of n, a(n) for n = 1..300</a>
%F Given g.f. S(x) = Sum_{n>=1} a(n)*x^(2*n) and related functions C(x) and F(x) (g.f. of A357787) satisfy the following.
%F (1) C(x)^2 + S(x)^2 = 1.
%F (2) C(x) + i*S(x) = Sum_{n=-oo..+oo} i^n * (2*x)^(n^2) * F(x)^n.
%F (3) 1/F(x) = F(-x).
%F (4) C(x) = 1 + Sum_{n>=1} (-1)^n * (2*x)^(4*n^2) * (F(x)^(2*n) + F(-x)^(2*n)).
%F (5) S(x) = Sum_{n>=0} (-1)^n * (2*x)^((2*n+1)^2) * (F(x)^(2*n+1) - F(-x)^(2*n+1)).
%F (6) C(x) + i*S(x) = Product_{n>=1} (1 + i*(2*x)^(2*n-1)*F(x)) * (1 - i*(2*x)^(2*n-1)/F(x)) * (1 - (2*x)^(2*n)), due to the Jacobi triple product identity.
%e G.f.: S(x) = 8*x^2 + 32*x^4 + 128*x^6 - 9216*x^10 - 94208*x^12 - 671744*x^14 - 3014656*x^16 + 1245184*x^18 + 171704320*x^20 + 1756364800*x^22 + 8338276352*x^24 - 26013073408*x^26 - 946201427968*x^28 - 10033714692096*x^30 - 56471303749632*x^32 + ...
%e such that S(x) and C(x) = sqrt(1 - S(x)^2) form the real and imaginary parts of the doubly infinite sum
%e C(x) + i*S(x) = Sum_{n=-oo..+oo} i^n * (2*x)^(n^2) * F(x)^n
%e where F(x) is the g.f. of A357787 and normalizes the given theta series so that C(x)^2 + S(x)^2 = 1.
%e Explicitly,
%e C(x) = 1 - (2*x)^4*(F(x)^2 + F(-x)^2) + (2*x)^16*(F(x)^4 + F(-x)^4) - (2*x)^36*(F(x)^6 + F(-x)^6) + (2*x)^64*(F(x)^8 + F(-x)^8) + ... + (-1)^n * (2*x)^(4*n^2) * (F(x)^(2*n) + F(-x)^(2*n)) + ...
%e S(x) = (2*x)*(F(x) - F(-x)) - (2*x)^9*(F(x)^3 - F(-x)^3) + (2*x)^25*(F(x)^5 - F(-x)^5) - (2*x)^49*(F(x)^7 - F(-x)^7) + ... + (-1)^n * (2*x)^((2*n+1)^2) * (F(x)^(2*n+1) - F(-x)^(2*n+1)) + ...
%e where
%e F(x) = 1 + 2*x + 2*x^2 + 8*x^3 + 14*x^4 + 32*x^5 + 68*x^6 + 22*x^8 - 768*x^9 - 2020*x^10 - 9216*x^11 - 23156*x^12 - 45056*x^13 - 115320*x^14 + 32768*x^15 + ... + A357787(n) * x^n + ...
%e C(x) = 1 - 32*x^4 - 256*x^6 - 2048*x^8 - 12288*x^10 - 32768*x^12 + 131072*x^14 + 3276800*x^16 + 28311552*x^18 + 125829120*x^20 - 285212672*x^22 - 11274289152*x^24 + ... + A357788(n)*x^(2*n) + ...
%o (PARI) {a(n) = my(F=[1,2],THETA=1); for(i=1,2*n, F = concat(F,0); m=sqrtint(#F+9);
%o THETA = sum(n=-m,m, I^n * (2*x)^(n^2) * truncate(Ser(F))^n + x*O(x^(#F+2)));
%o F[#F] = -polcoeff( (real(THETA)^2 + imag(THETA)^2)/64, #F+2)); polcoeff(imag(THETA),2*n)}
%o for(n=1,25,print1(a(n),", "))
%Y Cf. A357787, A357788.
%K sign
%O 1,1
%A _Paul D. Hanna_, Dec 05 2022
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