A277000 - OEIS

%I #24 Sep 09 2018 09:23:03

%S 1,-1,19,-2561,874831,-319094777,47095708213409,-751163826506551,

%T 281559662236405100437,-49061598325832137241324057,

%U 5012066724315488368700829665081,-26602063280041700132088988446735433,40762630349420684160007591156102493590477

%N Numerators of an asymptotic series for the Gamma function (even power series).

%C Let y = x+1/2 then Gamma(x+1) ~ sqrt(2*Pi)*((y/E)*Sum_{k>=0} r(k)/y^(2*k))^y as x -> oo and r(k) = A277000(k)/A277001(k) (see example 6.1 in the Wang reference).

%H Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/FactorialFunction">Approximations to the factorial function</a>.

%H 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 a(n) = numerator(b(2*n)) with b(n) = Y_{n}(0, z_2, z_3,..., z_n)/n! with z_k = k!*Bernoulli(k,1/2)/(k*(k-1)) and Y_{n} the complete Bell polynomials.

%F The rational numbers have the recurrence r(n) = (1/(2*n))*Sum_{m=0..n-1} Bernoulli(2*m+2,1/2)*r(n-m-1)/(2*m+1)) for n>=1, r(0)=1. - _Peter Luschny_, Sep 30 2016

%e The underlying rational sequence starts:

%e 1, 0, -1/24, 0, 19/5760, 0, -2561/2903040, 0, 874831/1393459200, 0, ...

%p b := n -> CompleteBellB(n, 0, seq((k-2)!*bernoulli(k,1/2), k=2..n))/n!:

%p A277000 := n -> numer(b(2*n)): seq(A277000(n), n=0..12);

%p # Alternatively the rational sequence by recurrence:

%p R := proc(n) option remember; local k; `if`(n=0, 1,

%p add(bernoulli(2*m+2,1/2)* R(n-m-1)/(2*m+1), m=0..n-1)/(2*n)) end:

%p seq(numer(R(n)), n=0..12); # _Peter Luschny_, Sep 30 2016

%t CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];

%t b[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1/2], {k, 2, n}]]]/n!;

%t a[n_] := Numerator[b[2n]];

%t Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Sep 09 2018 *)

%Y Cf. A001163/A001164 (Stirling), A182935/A144618 (De Moivre), A005146/A005147 (Stieltjes), A090674/A090675 (Lanczos), A181855/A181856 (Nemes), A182912/A182913 (NemesG), A182916/A182917 (Wehmeier), A182919/A182920 (Gosper), A182914/A182915, A277002/A277003 (odd power series).

%Y Cf. A276667/A276668 (the arguments of the Bell polynomials).

%K sign,frac

%O 0,3

%A _Peter Luschny_, Sep 25 2016