A355872 G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).
2, 14, 434, 17662, 829314, 42293582, 2276970482, 127359871870, 7328894334338, 431089922960910, 25803242957983410, 1566580082112919422, 96239944539571023362, 5971465584401568096846, 373681955307631772312050, 23556948108319423559281918, 1494606013410312933197468930
Offset: 1
Keywords
Examples
G.f. A(x) = 2*x + 14*x^5 + 434*x^9 + 17662*x^13 + 829314*x^17 + 42293582*x^21 + 2276970482*x^25 + 127359871870*x^29 + 7328894334338*x^33 + 431089922960910*x^37 + ... such that A = A(x) satisfies x = ... + 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) + ...
Links
- Paul D. Hanna, Table of n, a(n) for n = 1..300
Programs
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Mathematica
(* Calculation of constant d: *) 1/r^4 /. FindRoot[{r*s^4*QPochhammer[1/(r*s^3), r^2*s^2] * QPochhammer[s/r, r^2*s^2]*QPochhammer[r^2*s^2, r^2*s^2] == (r - s)*(-1 + r*s^3), 1/s^3*(3*s + r*(-4 + r*s^3) + 2*r^2*(r - s)*s^2*(-1 + r*s^3)* Derivative[0, 1][QPochhammer][1/(r*s^3), r^2*s^2] / QPochhammer[1/(r*s^3), r^2*s^2] + 2*r^3*s^6*QPochhammer[1/(r*s^3), r^2*s^2] * QPochhammer[r^2*s^2, r^2*s^2] * Derivative[0, 1][QPochhammer][s/r, r^2*s^2] + (r - s)*(-1 + r*s^3)* (-(2*QPolyGamma[0, 1, r^2*s^2] - 3*QPolyGamma[0, Log[1/(r*s^3)]/Log[r^2*s^2], r^2*s^2] + QPolyGamma[0, Log[s/r]/Log[r^2*s^2], r^2*s^2]) / Log[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}, {r, 1/60}, {s, 2}, WorkingPrecision -> 70] (* Vaclav Kotesovec, Jan 18 2024 *) -
PARI
{a(n) = my(A=[0,2]); for(i=1,n, A=concat(A,[0,0,0,0]); A[#A] = -polcoeff( sum(m=-#A,#A,(-x)^(m^2) * Ser(A)^((m-1)^2) ), #A-1)); A[4*n-2]} for(n=1,20,print1(a(n),", "))
Formula
G.f. A(x) = Sum_{n>=1} a(n) * x^(4*n-3) satisfies:
(1) x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n+1)^2).
(2) x = 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) -1 = 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 = 69.7705416198088434764685861402300375255728007801297265... and c = 0.0044667602848752470638241640199049506066862963974858... - Vaclav Kotesovec, Mar 19 2023
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