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A277000 Numerators of an asymptotic series for the Gamma function (even power series).

%I A277000 #30 Mar 07 2025 11:40:27
%S A277000 1,-1,19,-2561,874831,-319094777,47095708213409,-751163826506551,
%T A277000 281559662236405100437,-49061598325832137241324057,
%U A277000 5012066724315488368700829665081,-26602063280041700132088988446735433,40762630349420684160007591156102493590477
%N A277000 Numerators of an asymptotic series for the Gamma function (even power series).
%C A277000 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 A277000 Peter Luschny, <a href="https://oeis.org/wiki/User:Peter_Luschny/FactorialFunction">Approximations to the factorial function</a>.
%H A277000 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 A277000 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 A277000 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 A277000 The underlying rational sequence starts:
%e A277000 1, 0, -1/24, 0, 19/5760, 0, -2561/2903040, 0, 874831/1393459200, 0, ...
%p A277000 b := n -> CompleteBellB(n, 0, seq((k-2)!*bernoulli(k,1/2), k=2..n))/n!:
%p A277000 A277000 := n -> numer(b(2*n)): seq(A277000(n), n=0..12);
%p A277000 # Alternatively the rational sequence by recurrence:
%p A277000 R := proc(n) option remember; local k; `if`(n=0, 1,
%p A277000 add(bernoulli(2*m+2,1/2)* R(n-m-1)/(2*m+1), m=0..n-1)/(2*n)) end:
%p A277000 seq(numer(R(n)), n=0..12); # _Peter Luschny_, Sep 30 2016
%t A277000 CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}];
%t A277000 b[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1/2], {k, 2, n}]]]/n!;
%t A277000 a[n_] := Numerator[b[2n]];
%t A277000 Table[a[n], {n, 0, 12}] (* _Jean-François Alcover_, Sep 09 2018 *)
%Y A277000 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 A277000 Cf. A276667/A276668 (the arguments of the Bell polynomials).
%K A277000 sign,frac
%O A277000 0,3
%A A277000 _Peter Luschny_, Sep 25 2016