A144622 -id:A144622 - cp's OEIS frontend

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

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

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.

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

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).

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

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

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 *)

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

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

A144617 Triangle read by rows: numerators of coefficients of the Debye-type polynomial u_n used for asymptotic Airy-type expansions of Bessel functions of arbitrary large order.

Original entry on oeis.org

1, 3, -5, 81, -462, 385, 30375, -369603, 765765, -425425, 4465125, -94121676, 349922430, -446185740, 185910725, 1519035525, -49286948607, 284499769554, -614135872350, 566098157625, -188699385875

Offset: 0

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

			The polynomials u_0, u_1, u_2 and u_3 are:
1;
(3*t - 5*t^3)/24;
(81*t^2 - 462*t^4 + 385*t^6)/1152;
(30375*t^3 - 369603*t^5 + 765765*t^7 - 425425*t^9)/414720.

Chris Kormanyos, Rows n=0..121 of triangle, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972. See Section 9.3.9.
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]. See Section 9.3.9.

For denominators see A144618. Cf. A144622.

uktop = {1, 3, -5}; ukbot = {1, 24}; u = ((3 t) - (5 (t^3)))/24; Do[uk = (((1/2) (t^2) (1 - (t^2))) D[u, t]) + ((1/8) Integrate[((1 - (5 (t^2))) u), {t, 0, t}]); u = Simplify[uk]; Do[uktop = Append[uktop, Coefficient[Expand[Numerator[u]], t^n]], {n, k, 3 k, 2}]; ukbot = Append[ukbot, Denominator[u]]; Print[k], {k, 2, 8}]; (* Chris Kormanyos (ckormanyos(AT)yahoo.com), Jan 18 2009 *)

Terms up to u_5 from Chris Kormanyos (ckormanyos(AT)yahoo.com), Jan 18 2009

cp's OEIS Frontend

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

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