A144618 Denominators of an asymptotic series for the factorial function (Stirling's formula with half-shift).

1, 24, 1152, 414720, 39813120, 6688604160, 4815794995200, 115579079884800, 22191183337881600, 263631258054033408000, 88580102706155225088000, 27636992044320430227456000, 39797268543821419527536640000

Offset

0,2

Comments

From Peter Luschny, Feb 24 2011 (Start):

G_n = A182935(n)/A144618(n). These rational numbers provide the coefficients for an asymptotic expansion of the factorial function.

The relationship between these coefficients and the Bernoulli numbers are due to De Moivre, 1730 (see Laurie). (End)

Also denominators of polynomials mentioned in A144617.

Also denominators of polynomials mentioned in A144622.

Links

Chris Kormanyos, Table denominators of u_k for k=0..121

Dirk Laurie, Old and new ways of computing the gamma function, page 14, 2005.

Peter Luschny, Approximation Formulas for the Factorial Function.

W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).

Formula

z! ~ sqrt(2 Pi) (z+1/2)^(z+1/2) e^(-z-1/2) Sum_{n>=0} G_n / (z+1/2)^n.

- Peter Luschny, Feb 24 2011

Example

G_0 = 1, G_1 = -1/24, G_2 = 1/1152, G_3 = 1003/414720.

Maple

G := proc(n) option remember; local j, R;
R := seq(2*j, j=1..iquo(n+1, 2));
`if`(n=0, 1, add(bernoulli(j, 1/2)*G(n-j+1)/(n*j), j=R)) end:
A144618 := n -> denom(G(n)); seq(A144618(i), i=0..12);
# Peter Luschny, Feb 24 2011

Mathematica

a[0] = 1; a[n_] := a[n] = Sum[ BernoulliB[j, 1/2]*a[n-j+1]/(n*j), {j, 2, n+1, 2}]; Table[a[n] // Denominator, {n, 0, 12}] (* Jean-François Alcover, Jul 26 2013, after Maple *)

Crossrefs

Cf. A001163, A001164, A182935, A144617, A144622.

Sequence in context: A191744 A323994 A090675 * A042107 A042104 A069991

Adjacent sequences: A144615 A144616 A144617 * A144619 A144620 A144621

Keyword

nonn,frac

Author

N. J. A. Sloane, Jan 15 2009, based on email from Chris Kormanyos (ckormanyos(AT)yahoo.com)

Extensions

Added more terms up to polynomial number u_12, v_12 for the denominators of u_k, v_k. Christopher Kormanyos (ckormanyos(AT)yahoo.com), Jan 31 2009

Typo in definition corrected Aug 05 2010 by N. J. A. Sloane

A-number in definition corrected - R. J. Mathar, Aug 05 2010

Edited and new definition by Peter Luschny, Feb 24 2011

Status

approved