A366022 Decimal expansion of a constant related to the asymptotics of A109085.

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

Offset: 0

Author

Vaclav Kotesovec, Sep 26 2023

Keywords

• nonn
• cons

Examples

			0.489635226684303373081541660578468619322416625...
		

Crossrefs

Cf. A109085, A270915, A366018.

Programs

Mathematica

val = -s*Log[r*s] / Sqrt[2*Pi*((-2 - 3*Log[r*s] + 2*Log[1 - r*s])* QPolyGamma[0, 1, r*s] + QPolyGamma[0, 1, r*s]^2 - 4*ArcTanh[1 - 2*r*s]*(Log[r*s] - Log[1 - r*s]/2 - r*(s/(1 - r*s))) - 2*(Log[1 - r*s]/(1 - r*s)) - QPolyGamma[1, 1, r*s] + r*s*Log[r* s]*((-r)*s^2*Log[r*s]* Derivative[0, 2][QPochhammer][r*s, r*s] + 2*Derivative[0, 0, 1][QPolyGamma][0, 1, r*s]))] /. FindRoot[{s == 1/QPochhammer[r*s], 1/s + r*s*Derivative[0, 1][QPochhammer][r*s, r*s] == (Log[1 - r*s] + QPolyGamma[0, 1, r*s])/(s* Log[r*s])}, {r, 1/5}, {s, 1}, WorkingPrecision -> 1000]; RealDigits[Chop[val], 10, -Floor[Log[10, Abs[Im[val]]]] - 3][[1]]

Formula

Equals limit_{n->infinity} A109085(n) * n^(3/2) / A270915^n.