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 A357803 a(n) = coefficient of x^(2*n) in A(x) such that A(x) = G(x)^2 where G(x) = 1 + Sum_{n>=1} (-1)^n * x^(4*n^2) * (F(x/2)^(2*n) + F(-x/2)^(2*n)), and F(x) is the g.f. of A357787. 1
 1, 0, -4, -8, -12, -8, 32, 128, 292, 440, 248, -904, -3616, -7032, -5824, 13056, 66372, 146144, 145116, -250216, -1545848, -3862464, -5072992, 2525376, 33963072, 100587096, 164044888, 37400880, -712627392, -2586929240, -5043112640, -3653527040, 13772747140, 65564551200, 150393547384, 167883535904 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Equals the self-convolution of A357806. LINKS Paul D. Hanna, Table of n, a(n) for n = 0..300 FORMULA G.f. A(x) = Sum_{n>=0} a(n)*x^(2*n) and related function F(x) (g.f. of A357787) satisfy the following. (1) A(x) = ( 1 + Sum_{n>=1} (-1)^n * x^(4*n^2) * (F(x/2)^(2*n) + F(-x/2)^(2*n)) )^2. (2) sqrt(A(x)) + i*sqrt(1 - A(x)) = Sum_{n=-oo..+oo} i^n * x^(n^2) * F(x/2)^n. (3) sqrt(A(x)) + i*sqrt(1 - A(x)) = 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. EXAMPLE G.f.: A(x) = 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 + ... where sqrt(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 + ... + A357806(n)*x^(2*n) + ... The related function F(x) is the g.f. of A357787 and begins 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 + ... + A357787(n)*x^n + ... where sqrt(A(x)) = 1 + Sum_{n>=1} (-1)^n * x^(4*n^2) * (F(x/2)^(2*n) + F(-x/2)^(2*n)). PROG (PARI) {a(n) = my(F=[1, 2], THETA=1); for(i=1, 2*n, F = concat(F, 0); m=sqrtint(#F+9); THETA = sum(n=-m, m, I^n * (2*x)^(n^2) * truncate(Ser(F))^n + x*O(x^(#F+2))); F[#F] = -polcoeff( (real(THETA)^2 + imag(THETA)^2)/64, #F+2)); polcoeff(real(THETA)^2, 2*n)/4^n} for(n=0, 35, print1(a(n), ", ")) CROSSREFS Cf. A357806, A357787, A357788, A357789. Sequence in context: A308653 A196268 A251756 * A158457 A072905 A086481 Adjacent sequences: A357800 A357801 A357802 * A357804 A357805 A357806 KEYWORD sign AUTHOR Paul D. Hanna, Dec 06 2022 STATUS approved

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Last modified August 4 17:39 EDT 2024. Contains 374923 sequences. (Running on oeis4.)