A246948 Decimal expansion of the coefficient c appearing in the asymptotic expression of the probability that a random n-permutation is a cube as c/n^3.

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

Offset: 1

Author

Jean-François Alcover, Sep 08 2014

Keywords

• nonn
• cons

Examples

			1.072997944389527017737971394954465555681...
		

Links

• Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 36.

Crossrefs

Cf. A246945.

Programs

Maple

evalf(3^(5/6) * GAMMA(1/3) / (2*Pi) * Product(1/3*(exp(1/(3*k)) + 2*exp(-1/(6*k)) * cos(sqrt(3)/(6*k))), k=1..infinity),100) # Vaclav Kotesovec, Sep 17 2014

Mathematica

digits = 40; m0 = 1000; dm = 1000; psi[x_] := 1/3*(E^x + 2*E^(-x/2)*Cos[Sqrt[3]*(x/2)]); tail[m_] := (-98761420800*PolyGamma[2, m] - 4572288*PolyGamma[5, m] - 53*PolyGamma[8, m])/31998700339200; Clear[f]; f[m_] := f[m] = Sum[Log[psi[1/(3*k)]], {k, 1, m - 1}] + tail[m] // N[#, digits + 10] &; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits + 5] != RealDigits[f[m - dm], 10, digits + 5], Print["f(", m, ") = ", f[m]]; m = m + dm]; c = 3^(5/6)*Gamma[1/3]/(2*Pi)*E^f[m]; RealDigits[c, 10, 40] // First

PARI

default(realprecision,150); 3^(5/6) * gamma(1/3) / (2*Pi) * exp(sumpos(k=1,log(1/3*(exp(1/(3*k)) + 2*exp(-1/(6*k)) * cos(sqrt(3)/(6*k)))))) \\ Vaclav Kotesovec, Sep 21 2014

Formula

c = 3^(5/6)*Gamma(1/3)/(2*Pi)*prod_{k>=1} psi(1/(3k)), where psi(x) = 1/3*(e^x + 2*e^(-x/2)*cos(sqrt(3)*(x/2))).

Extensions

More terms from Vaclav Kotesovec, Sep 17 2014