cp's OEIS Frontend

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).

%I A246945 #39 Dec 28 2020 07:27:35
%S A246945 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,
%T A246945 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,
%U A246945 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
%N 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).
%D A246945 See A003483.
%H A246945 Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, p. 41.
%H A246945 Ph. Flajolet, É. Fusy, X. Gourdon, D. Panario, N. Pouyanne, <a href="http://arxiv.org/abs/math/0606370">A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics</a>, arXiv:math/0606370 [math.CO]
%F A246945 e^G = prod_{k>=1} cosh(1/(2k)).
%F A246945 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
%e A246945 G = 0.2003084150040401276417752235643787366634879653405876198956293474890714...
%e A246945 e^G = 1.22177951519253683396485298445636121278881014814697728683863962970923...
%e A246945 sqrt(2/Pi)*e^G = 0.974839011877335012323657925154410019528043463671159620094...
%p A246945 evalf(1/(product(sech(1/(2*k)), k=1..infinity)), 120) # _Vaclav Kotesovec_, Sep 20 2014
%t A246945 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
%t A246945 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 *)
%Y A246945 Cf. A003483, A249673.
%K A246945 nonn,cons
%O A246945 1,2
%A A246945 _Jean-François Alcover_, Sep 08 2014
%E A246945 More terms from _Vaclav Kotesovec_, Sep 20 2014