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Decimal expansion of a constant related to A008485 and A327215.
4

%I #7 Oct 02 2023 13:09:47

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

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

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

%N Decimal expansion of a constant related to A008485 and A327215.

%F Equals limit_{n->infinity} A008485(n) * sqrt(n) / A270915^n.

%e 0.26801521271073331568695383828032867950066675724203942644590415846953909499267...

%t val = Sqrt[(1 - r*s)*(Log[r*s]^2/(2*Pi*(4*ArcTanh[1 - 2*r*s]*(r*s + (-1 + r*s)*Log[r*s]) - 2*(1 + (-1 + r*s)*ArcTanh[1 - 2*r*s])*Log[1 - r*s] + (-1 + r*s)*(2 + 3*Log[r*s] - 2*Log[1 - r*s]) * QPolyGamma[0, 1, r*s] + (1 - r*s)* QPolyGamma[0, 1, r*s]^2 + (-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[{QPochhammer[r*s] == 1/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, 2}, WorkingPrecision -> 1000]; RealDigits[Chop[val], 10, -Floor[Log[10, Abs[Im[val]]]] - 3][[1]] (* _Vaclav Kotesovec_, Oct 02 2023 *)

%Y Cf. A008485, A270915, A327215, A327280, A366022.

%K nonn,cons

%O 0,1

%A _Vaclav Kotesovec_, Aug 28 2019