OFFSET
0,4
COMMENTS
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).
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
KEYWORD
sign,frac
AUTHOR
Peter Luschny, Dec 02 2010
STATUS
approved