A366174 - OEIS

%I #5 Oct 03 2023 04:19:43

%S 4,9,8,3,3,4,7,9,7,9,3,3,6,0,3,4,2,2,6,0,6,3,5,9,2,6,4,0,2,8,5,0,0,1,

%T 6,4,4,3,0,6,9,4,2,8,2,3,3,0,5,1,2,9,0,2,0,1,9,9,6,8,5,3,4,9,8,3,4,0,

%U 8,7,7,4,8,5,5,2,2,7,8,4,0,2,8,9,1,1,4,7,7,3,5,2,9,2,4,3,0,3,5,0,6,2,6,0,3,1

%N Decimal expansion of a constant related to the asymptotics of A106336.

%F Equals limit_{n->oo} A106336(n) * n^(3/2) * A106335^n.

%e 0.4983347979336034226063592640285001644306942823305129020199685349834...

%t val = Sqrt[(s^2*(-1 + r^2*s^2)*Log[r^2*s^2]^2 * QPochhammer[-1, r*s]^2)/ (2*Pi*(-2*r^2*s^2*(-1 + r^2*s^2)*Log[r^2*s^2]^2* Derivative[0, 1][QPochhammer][-1, r*s]^2 + r*s*(-1 + r^2*s^2)*Log[r^2*s^2]*QPochhammer[-1, r*s]* ((Log[r^2*s^2] + 4*Log[1 - r^2*s^2] + 4*QPolyGamma[0, 1, r^2*s^2])* Derivative[0, 1][QPochhammer][-1, r*s] + r*s*Log[r^2*s^2] * Derivative[0, 2][QPochhammer][-1, r*s]) + 2*r^4*s^3*(-1 + r^2*s^2) * Log[r^2*s^2]^2*QPochhammer[-1, r*s]^3 * Derivative[0, 2][QPochhammer][r^2*s^2, r^2*s^2] + QPochhammer[-1, r*s]^2*(16*r^2*s^2*ArcTanh[1 - 2*r^2*s^2] + (-1 + r^2*s^2)*Log[r^2*s^2]^2 - 8*Log[1 - r^2*s^2] - 4*(-1 + r^2*s^2)* Log[1 - r^2*s^2]^2 - 8*(-1 + r^2*s^2)*(-1 + Log[1 - r^2*s^2])* QPolyGamma[0, 1, r^2*s^2] + (4 - 4*r^2*s^2)* QPolyGamma[0, 1, r^2*s^2]^2 + 4*(-1 + r^2*s^2)*(QPolyGamma[1, 1, r^2*s^2] - 2*r^2*s^2*Log[r^2*s^2]* Derivative[0, 0, 1][QPolyGamma][0, 1, r^2*s^2]))))] /. FindRoot[{2*s == QPochhammer[-1, r*s]*QPochhammer[r^2*s^2], (-2*Log[1 - r^2*s^2] - 2*QPolyGamma[0, 1, r^2*s^2])/ Log[r^2*s^2] + (r* s*(Derivative[0, 1][QPochhammer][-1, r*s] + r*QPochhammer[-1, r*s]^2* Derivative[0, 1][QPochhammer][r^2*s^2, r^2*s^2]))/ QPochhammer[-1, r*s] == 1}, {r, 1/3}, {s, 2}, WorkingPrecision -> 600]; RealDigits[Chop[val], 10, -Floor[Log[10, Abs[Im[val]]]] - 3][[1]] (* r = A106335, s = A106334 *)

%Y Cf. A106334, A106335, A106336.

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, Oct 03 2023