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A181856 Denominator of Nemes number G_n. 4
1, 12, 1440, 362880, 87091200, 11496038400, 376610217984000, 903864523161600, 36877672544993280000, 529710888436283473920000, 3496091863679470927872000000, 50785334440817577689088000000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

G(n) = A181855(n)/A181856(n). Nemes numbers provide the coefficients for an asymptotic expansion for the Gamma function for real arguments greater or equal than one.

Gamma(x) = sqrt(2Pi/x)((x/e)(Sum_{0<=k<n} G_k x^(-2k) + R_n(x)))^x.

LINKS

Table of n, a(n) for n=0..11.

Gergő Nemes, New asymptotic expansion for the Gamma function, Arch. Math. 95 (2010), 161-169, Springer Basel.

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

FORMULA

G_0 = 1 and for n > 1 and B_n denoting the Bernoulli number,

G_n = Sum_{m=0..n} B_{2m+2} G_{n-m-1} / (2m+1),m=0..n-1)/(2n)).

a(n) = denominator(p(2*n)) with p(n) = Y_{n}(0, z_2, z_3,..., z_n)/n! with z_k = (k-2)!*Bernoulli(k,1) and Y_{n} the complete Bell polynomials. - Peter Luschny, Oct 03 2016

EXAMPLE

G_0 = 1, G_1 = 1/12, G_2 = 1/1440, G_3 = 239/362880.

MAPLE

G := proc(n) option remember; local k; `if`(n=0, 1,

add(bernoulli(2*m+2)*G(n-m-1)/(2*m+1), m=0..n-1)/(2*n)) end;

a181856 := n -> denom(G(n));

MATHEMATICA

a[0] = 1; a[n_] := a[n] = Sum[ BernoulliB[2m + 2]*a[n - m - 1]/(2m + 1), {m, 0, n}]/(2n); Table[a[n] // Denominator, {n, 0, 11}] (* Jean-François Alcover, Jul 26 2013 *)

CROSSREFS

Cf. A000367, A002445, A181855 (numerators).

Sequence in context: A008992 A260448 A271514 * A161149 A160490 A276905

Adjacent sequences:  A181853 A181854 A181855 * A181857 A181858 A181859

KEYWORD

nonn,frac

AUTHOR

Peter Luschny, Dec 02 2010

STATUS

approved

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Last modified September 22 00:25 EDT 2017. Contains 292326 sequences.