A181855 Numerator of Nemes numbers G_n.

1, 1, 1, 239, -46409, 9113897, -695818219549, 5649766313929, -1070083202835456443, 93856597276403726428217, -4815785492460413153189484781, 674781102986061046417681986493, -9845646538265462155478818981872958283

Offset

0,4

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.

Links

Table of n, a(n) for n=0..12.

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).

Formula

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

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

a(n) = numerator(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

Example

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

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;
a181855 := n -> numer(G(n));
# Alternatively:
p := n -> CompleteBellB(n, 0, seq((k-2)!*bernoulli(k, 1), k=2..n))/n!:
a := n -> numer(p(2*n)): seq(a(n), n=0..12); # Peter Luschny, Oct 03 2016

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] // Numerator, {n, 0, 12}] (* 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_] := Numerator[p[2n]];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Sep 09 2018 *)

Crossrefs

Cf. A000367, A002445, A181856 (denominators).

Sequence in context: A069364 A163052 A254298 * A223741 A223724 A223788

Adjacent sequences: A181852 A181853 A181854 * A181856 A181857 A181858

Keyword

sign,frac

Author

Peter Luschny, Dec 02 2010

Status

approved