A277000 Numerators of an asymptotic series for the Gamma function (even power series).
1, -1, 19, -2561, 874831, -319094777, 47095708213409, -751163826506551, 281559662236405100437, -49061598325832137241324057, 5012066724315488368700829665081, -26602063280041700132088988446735433, 40762630349420684160007591156102493590477
Offset: 0
Examples
The underlying rational sequence starts: 1, 0, -1/24, 0, 19/5760, 0, -2561/2903040, 0, 874831/1393459200, 0, ...
Links
- Peter Luschny, Approximations to the factorial function.
- W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).
Crossrefs
Programs
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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
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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 *)
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
Comments