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A181856 Denominator of Nemes number G_n.

%I A181856 #28 Apr 15 2023 06:23:30
%S A181856 1,12,1440,362880,87091200,11496038400,376610217984000,
%T A181856 903864523161600,36877672544993280000,529710888436283473920000,
%U A181856 3496091863679470927872000000,50785334440817577689088000000
%N A181856 Denominator of Nemes number G_n.
%C A181856 G(n) = A181855(n)/A181856(n). Nemes numbers provide the coefficients for an asymptotic expansion for the Gamma function for real arguments greater than or equal to one.
%C A181856 Gamma(x) = sqrt(2*Pi/x)*((x/e)*(Sum_{k=0..n-1} G_k x^(-2k) + R_n(x)))^x.
%H A181856 Gergő Nemes, <a href="http://dx.doi.org/10.1007/s00013-010-0146-9">New asymptotic expansion for the Gamma function</a>, Arch. Math. 95 (2010), 161-169, Springer Basel.
%H A181856 W. Wang, <a href="http://dx.doi.org/10.1016/j.jnt.2015.12.016">Unified approaches to the approximations of the gamma function</a>, J. Number Theory (2016).
%F A181856 G_0 = 1 and for n > 1 and B_n denoting the Bernoulli number, we have
%F A181856 G_n = Sum_{m=0..n} B_{2m+2} * G_{n-m-1} / ((2m+1) * (2*n)).
%F A181856 a(n) = denominator(p(2*n)) with p(n) = Y_{n}(0, z_2, z_3, ..., z_n)/n! with z_k = (k-2)!*Bernoulli(k,1) and Y_{n} the complete Bell polynomials. - _Peter Luschny_, Oct 03 2016
%e A181856 G_0 = 1, G_1 = 1/12, G_2 = 1/1440, G_3 = 239/362880.
%p A181856 G := proc(n) option remember; local k; `if`(n=0,1,
%p A181856 add(bernoulli(2*m+2)*G(n-m-1)/(2*m+1),m=0..n-1)/(2*n)) end;
%p A181856 a181856 := n -> denom(G(n));
%t A181856 a[0] = 1;
%t A181856 a[n_] := a[n] = Sum[ BernoulliB[2m + 2]*a[n - m - 1]/(2m + 1), {m, 0, n}]/(2n);
%t A181856 Table[a[n] // Denominator, {n, 0, 11}] (* _Jean-François Alcover_, Jul 26 2013 *)
%t A181856 CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
%t A181856 p[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1], {k, 2, n}]]]/n!;
%t A181856 a[n_] := Denominator[p[2n]];
%t A181856 Table[a[n], {n, 0, 11}] (* _Jean-François Alcover_, Sep 09 2018 *)
%Y A181856 Cf. A000367, A002445, A181855 (numerators).
%K A181856 nonn,frac
%O A181856 0,2
%A A181856 _Peter Luschny_, Dec 02 2010