A277002 Numerators of an asymptotic series for the Gamma function (odd power series).

-1, 7, -31, 127, -511, 1414477, -8191, 118518239, -5749691557, 91546277357, -23273283019, 1982765468311237, -22076500342261, 455371239541065869, -925118910976041358111, 16555640865486520478399, -1302480594081611886641, 904185845619475242495834469891

Offset

1,2

Comments

Let y = x+1/2 then Gamma(x+1) ~ sqrt(2*Pi)*(y/E)^y*exp(Sum_{k>=1} r(k)/y^(2*k-1)) as x -> oo and r(k) = A277002(k)/A277003(k) (see example 7.1 in the Wang reference).

See also theorem 2 and formula (58) in Borwein and Corless. - Peter Luschny, Mar 31 2017

Links

Table of n, a(n) for n=1..18.

J. M. Borwein, R. M. Corless, Gamma and Factorial in the Monthly, arXiv:1703.05349 [math.HO], 2017.

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-1)) with b(n) = Bernoulli(n+1, 1/2)/(n*(n+1)) for n>=1, b(0)=0.

Example

The underlying rational sequence b(n) starts:
0, -1/24, 0, 7/2880, 0, -31/40320, 0, 127/215040, 0, -511/608256, ...

Maple

b := n -> `if`(n=0, 0, bernoulli(n+1, 1/2)/(n*(n+1))):
a := n -> numer(b(2*n-1)):
seq(a(n), n=1..18);

Mathematica

b[n_] := BernoulliB[n+1, 1/2]/(n(n+1));
a[n_] := Numerator[b[2n-1]];
Array[a, 18] (* Jean-François Alcover, Sep 09 2018 *)

Crossrefs

Cf. A277003 (denominators), A277000/A277001 (even power series).

Sequence in context: A255282 A303449 A083420 * A282898 A036282 A033474

Adjacent sequences: A276999 A277000 A277001 * A277003 A277004 A277005

Keyword

sign,frac

Author

Peter Luschny, Sep 26 2016

Status

approved