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Denominators in Stirling's asymptotic series.
2

%I #15 Sep 23 2022 17:20:13

%S 1,12,144,8640,103680,1741824,104509440,179159040,2149908480,

%T 1418939596800,23838185226240,338068808663040,20284128519782400,

%U 18723810941337600,32097961613721600,229179445921972224000

%N Denominators in Stirling's asymptotic series.

%C Numerators coincide with the numbers depicted in A001163 but differ for the first time at entry nr. 33. See the W. Lang link.

%C Stirling's formula for Gamma(z) (|arg(z)| < Pi) uses the asymptotic series Sum_{k>=0} (N(k)/a(k))*((1/z)^k)/k!. For N(k) see the W. Lang link.

%H W. Lang, <a href="/A097303/a097303.txt">More terms and comments</a>.

%F a(n) = denominator(s(n)), where the signed rationals s(n) are the coefficients of ((1/z)^k)/k! in the asymptotic series appearing in Stirling's formula for Gamma(z).

%t max = 15; se = Series[(E^x*Sqrt[1/x]*Gamma[x+1])/(x^x*Sqrt[2*Pi]), {x, Infinity, max}]; Denominator[ CoefficientList[ se /. x -> 1/x, x]*Range[0, max]!] (* _Jean-François Alcover_, Nov 03 2011 *)

%Y Cf. A001163, A001164 (Stirling formula with further links and references.).

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Aug 13 2004