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 A277000 Numerators of an asymptotic series for the Gamma function (even power series). 5
 1, -1, 19, -2561, 874831, -319094777, 47095708213409, -751163826506551, 281559662236405100437, -49061598325832137241324057, 5012066724315488368700829665081, -26602063280041700132088988446735433, 40762630349420684160007591156102493590477 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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). LINKS Peter Luschny, Approximations to the factorial function. W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016). FORMULA 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. 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 EXAMPLE The underlying rational sequence starts: 1, 0, -1/24, 0, 19/5760, 0, -2561/2903040, 0, 874831/1393459200, 0, ... MAPLE b := n -> CompleteBellB(n, 0, seq((k-2)!*bernoulli(k, 1/2), k=2..n))/n!: A277000 := n -> numer(b(2*n)): seq(A277000(n), n=0..12); # Alternatively the rational sequence by recurrence: R := proc(n) option remember; local k; `if`(n=0, 1, add(bernoulli(2*m+2, 1/2)* R(n-m-1)/(2*m+1), m=0..n-1)/(2*n)) end: seq(numer(R(n)), n=0..12); # Peter Luschny, Sep 30 2016 MATHEMATICA CompleteBellB[n_, zz_] := Sum[BellY[n, k, zz[[1 ;; n-k+1]]], {k, 1, n}]; b[n_] := CompleteBellB[n, Join[{0}, Table[(k-2)! BernoulliB[k, 1/2], {k, 2, n}]]]/n!; a[n_] := Numerator[b[2n]]; Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Sep 09 2018 *) CROSSREFS 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). Cf. A276667/A276668 (the arguments of the Bell polynomials). Sequence in context: A178025 A172651 A183739 * A055415 A196541 A221296 Adjacent sequences:  A276997 A276998 A276999 * A277001 A277002 A277003 KEYWORD sign,frac AUTHOR Peter Luschny, Sep 25 2016 STATUS approved

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Last modified March 31 03:48 EDT 2020. Contains 333136 sequences. (Running on oeis4.)