A181856 Denominator of Nemes number G_n.

1, 12, 1440, 362880, 87091200, 11496038400, 376610217984000, 903864523161600, 36877672544993280000, 529710888436283473920000, 3496091863679470927872000000, 50785334440817577689088000000

Offset: 0

Author

Peter Luschny, Dec 02 2010

Keywords

• nonn
• frac

Comments

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.

Gamma(x) = sqrt(2*Pi/x)*((x/e)*(Sum_{k=0..n-1} G_k x^(-2k) + R_n(x)))^x.

Examples

			G_0 = 1, G_1 = 1/12, G_2 = 1/1440, G_3 = 239/362880.
		

Links

• Gergő Nemes, New asymptotic expansion for the Gamma function, Arch. Math. 95 (2010), 161-169, Springer Basel.
• W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).

Crossrefs

Cf. A000367, A002445, A181855 (numerators).

Programs

Maple

G := proc(n) option remember; local k; `if`(n=0,1,
add(bernoulli(2*m+2)*G(n-m-1)/(2*m+1),m=0..n-1)/(2*n)) end;
a181856 := n -> denom(G(n));

Mathematica

a[0] = 1;
a[n_] := a[n] = Sum[ BernoulliB[2m + 2]*a[n - m - 1]/(2m + 1), {m, 0, n}]/(2n);
Table[a[n] // Denominator, {n, 0, 11}] (* Jean-François Alcover, Jul 26 2013 *)
CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
p[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1], {k, 2, n}]]]/n!;
a[n_] := Denominator[p[2n]];
Table[a[n], {n, 0, 11}] (* Jean-François Alcover, Sep 09 2018 *)

Formula

G_0 = 1 and for n > 1 and B_n denoting the Bernoulli number, we have

G_n = Sum_{m=0..n} B_{2m+2} * G_{n-m-1} / ((2m+1) * (2*n)).

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