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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

Table of n, a(n) for n=0..12.

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

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 October 20 17:39 EDT 2017. Contains 293648 sequences.