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A366022
Decimal expansion of a constant related to the asymptotics of A109085.
2
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
0,1
FORMULA
Equals limit_{n->infinity} A109085(n) * n^(3/2) / A270915^n.
EXAMPLE
0.489635226684303373081541660578468619322416625...
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]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Sep 26 2023
STATUS
approved