A246945 Decimal expansion of the coefficient e^G appearing in the asymptotic expression of the probability that a random n-permutation is a square, as sqrt(2/Pi)*e^G/sqrt(n).
1, 2, 2, 1, 7, 7, 9, 5, 1, 5, 1, 9, 2, 5, 3, 6, 8, 3, 3, 9, 6, 4, 8, 5, 2, 9, 8, 4, 4, 5, 6, 3, 6, 1, 2, 1, 2, 7, 8, 8, 8, 1, 0, 1, 4, 8, 1, 4, 6, 9, 7, 7, 2, 8, 6, 8, 3, 8, 6, 3, 9, 6, 2, 9, 7, 0, 9, 2, 3, 3, 0, 4, 0, 3, 0, 0, 4, 8, 9, 3, 7, 3, 9, 9, 9, 6, 6, 2, 9, 8, 4, 3, 6, 7, 7, 8, 7, 9, 8, 7, 5, 8, 6, 7, 0
Offset: 1
Examples
G = 0.2003084150040401276417752235643787366634879653405876198956293474890714... e^G = 1.22177951519253683396485298445636121278881014814697728683863962970923... sqrt(2/Pi)*e^G = 0.974839011877335012323657925154410019528043463671159620094...
References
- See A003483.
Links
- Steven R. Finch, Errata and Addenda to Mathematical Constants, p. 41.
- Ph. Flajolet, É. Fusy, X. Gourdon, D. Panario, N. Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math/0606370 [math.CO]
Programs
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Maple
evalf(1/(product(sech(1/(2*k)), k=1..infinity)), 120) # Vaclav Kotesovec, Sep 20 2014
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Mathematica
digits = 42; m0 = 10^4; dm = 1000; tail[m_] := (406425600*PolyGamma[1, m] - 2822400*PolyGamma[3, m] + 9408*PolyGamma[5, m] - 17*PolyGamma[7, m])/3251404800; Clear[g]; g[m_] := g[m] = Sum[Log[Cosh[1/(2*k)]], {k, 1, m - 1}] + tail[m] // N[#, digits + 10] &; g[m0] ; g[m = m0 + dm]; While[RealDigits[g[m], 10, digits + 5] != RealDigits[g[m - dm], 10, digits + 5], Print["m = ", m]; m = m + dm]; G = g[m]; RealDigits[E^G, 10, digits ] // First Block[{$MaxExtraPrecision = 1000}, Do[Print[N[Exp[Sum[(-1)^(n + 1)*Zeta[2*n]^2*(1 - 1/2^(2*n))/n/Pi^(2*n), {n, 1, m}]], 120]], {m, 100, 150}]] (* Vaclav Kotesovec, Sep 20 2014 *)
Formula
e^G = prod_{k>=1} cosh(1/(2k)).
G = Sum_{n>=1} (-1)^(n+1) * Zeta(2*n)^2 * (1-1/2^(2*n)) / (n * Pi^(2*n)). - Vaclav Kotesovec, Sep 20 2014
Extensions
More terms from Vaclav Kotesovec, Sep 20 2014