A357806 - OEIS

%I #10 Dec 06 2022 10:23:06

%S 1,0,-2,-4,-8,-12,-8,8,50,108,120,-68,-672,-1644,-1904,1912,15456,

%T 41160,59494,-5852,-311040,-996744,-1752680,-840600,5988928,24181500,

%U 50438488,45910304,-103373216,-582387300,-1428882832,-1814475760,1263429058,13685575400

%N a(n) = coefficient of x^(2*n) in A(x) = 1 + Sum_{n>=1} (-1)^n * x^(4*n^2) * (F(x/2)^(2*n) + F(-x/2)^(2*n)), where F(x) is the g.f. of A357787.

%C a(n) = A357788(n)/4^n for n >= 0.

%H Paul D. Hanna, <a href="/A357806/b357806.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) and related function F(x) (g.f. of A357787) satisfy the following.

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

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

%F (3) A(x) + i*sqrt(1 - A(x)^2) = Product_{n>=1} (1 + i*x^(2*n-1)*F(x/2)) * (1 - i*x^(2*n-1)/F(x/2)) * (1 - x^(2*n)), due to the Jacobi triple product identity.

%e G.f.: A(x) = 1 - 2*x^4 - 4*x^6 - 8*x^8 - 12*x^10 - 8*x^12 + 8*x^14 + 50*x^16 + 108*x^18 + 120*x^20 - 68*x^22 - 672*x^24 - 1644*x^26 - 1904*x^28 + 1912*x^30 + 15456*x^32 + 41160*x^34 + 59494*x^36 - 5852*x^38 - 311040*x^40 + ... + A357788(n)*x^(2*n)/4^n + ...

%e The related function F(x) is the g.f. of A357787 and begins

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

%e The square of the g.f. A(x) begins:

%e A(x)^2 = 1 - 4*x^4 - 8*x^6 - 12*x^8 - 8*x^10 + 32*x^12 + 128*x^14 + 292*x^16 + 440*x^18 + 248*x^20 - 904*x^22 - 3616*x^24 - 7032*x^26 - 5824*x^28 + 13056*x^30 + 66372*x^32 + 146144*x^34 + 145116*x^36 - 250216*x^38 - 1545848*x^40 + ...

%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(real(THETA),2*n)/4^n}

%o for(n=0,35,print1(a(n),", "))

%Y Cf. A357787, A357788, A357789.

%K sign

%O 0,3

%A _Paul D. Hanna_, Dec 05 2022