OFFSET
0,1
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..300
FORMULA
G.f. A(x) satisfies:
(1) 2 = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n+1)^2).
(2) 2 = A(x) * Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n-1)*A(x)^(2*n+1)) * (1 - x^(2*n-1)*A(x)^(2*n-3)), by the Jacobi triple product identity.
(3) 2 = (-x) * Product_{n>=1} (1 - x^(2*n)*A(x)^(2*n)) * (1 - x^(2*n+1)*A(x)^(2*n-1)) * (1 - x^(2*n-3)*A(x)^(2*n-1)), by the Jacobi triple product identity.
a(n) ~ c * d^n / n^(3/2), where d = 77.309779325704779292317107617559471210592218708634855530355675234... and c = 0.31219183409397424726366930735250286274022579073644627976468... - Vaclav Kotesovec, Mar 19 2023
EXAMPLE
G.f.: A(x) = 2 + 17*x + 544*x^2 + 24344*x^3 + 1261702*x^4 + 71159152*x^5 + 4240009152*x^6 + 262584135640*x^7 + 16734002688722*x^8 + ...
such that A = A(x) satisfies
2 = ... + x^16*A^25 - x^9*A^16 + x^4*A^9 - x*A^4 + A - x + x^4*A - x^9*A^4 + x^16*A^9 - x^25*A^16 +- ... + (-x)^(n^2) * A(x,y)^((n-1)^2) + ...
MATHEMATICA
(* Calculation of constant d: *) 1/r /. FindRoot[{k == r^4*s^2 * QPochhammer[1/(r^3*s), r^2*s^2] * QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]/((r - s)*(-1 + r^3*s)), 1/r^3*(k*(1 + r^4 - 2*r/s) + 2*r^6*s^3*QPochhammer[r/s, r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2]* Derivative[0, 1][QPochhammer][1/(r^3*s), r^2*s^2] + 2*k*r^2*(r - s)*s*(-1 + r^3*s) * Derivative[0, 1][QPochhammer][r/s, r^2*s^2]/ QPochhammer[r/s, r^2*s^2] + 1/s*k*(r - s)*(-1 + r^3*s) * (1/ Log[r^2*s^2]*(-2*QPolyGamma[0, 1, r^2*s^2] + QPolyGamma[0, Log[1/(r^3*s)] / Log[r^2*s^2], r^2*s^2] + QPolyGamma[0, Log[r/s] / Log[r^2*s^2], r^2*s^2]) + 2*r^2*s^2 * Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2] / QPochhammer[r^2*s^2, r^2*s^2])) == 0} /. k -> 2, {r, 1/75}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 18 2024 *)
PROG
(PARI) {a(n) = my(A=[2], M); for(i=1, n, A = concat(A, 0); M = ceil(sqrt(n+1));
A[#A] = -polcoeff( sum(m=-M, M, (-x)^(m^2)*Ser(A)^((m-1)^2)), #A-1)); H=A; A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Aug 09 2022
STATUS
approved