%I #49 Feb 21 2022 02:16:12

%S 1,-1,1,-1,1,-691,1,-3617,43867,-174611,77683,-236364091,657931,

%T -3392780147,1723168255201,-7709321041217,151628697551,

%U -26315271553053477373,154210205991661,-261082718496449122051,1520097643918070802691

%N Numerators of coefficients in Stirling's expansion for log(Gamma(z)).

%C A001067(n) = a(n) if n<574; A001067(574) = 37*a(574). - _Michael Somos_, Feb 01 2004

%D M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.41.

%D L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205

%H Seiichi Manyama, <a href="/A046968/b046968.txt">Table of n, a(n) for n = 1..314</a>

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.41.

%H R. P. Brent, <a href="http://arxiv.org/abs/1608.04834"> Asymptotic approximation of central binomial coefficients with rigorous error bounds</a>, arXiv:1608.04834 [math.NA], 2016.

%H N. Elezovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Elezovic/elezovic5.html">Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers</a>, J. Int. Seq. 17 (2014) # 14.2.1.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StirlingsSeries.html">Stirling's Series</a>

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

%F From numerator of Jk(z) = (-1)^(k-1)*Bk/(((2k)*(2k-1))*z^(2k-1)), so Gamma(z) = sqrt(2*Pi)*z^(z-0.5)*exp(-z)*exp(J(z)).

%p seq(numer(bernoulli(2*n)/(2*n*(2*n-1))), n = 1..25); # _G. C. Greubel_, Sep 19 2019

%t Table[ Numerator[ BernoulliB[2n]/(2n(2n - 1))], {n, 1, 22}] (* _Robert G. Wilson v_, Feb 03 2004 *)

%t s = LogGamma[z] + z - (z - 1/2) Log[z] - Log[2 Pi]/2 + O[z, Infinity]^42; DeleteCases[CoefficientList[s, 1/z], 0] // Numerator (* _Jean-François Alcover_, Jun 13 2017 *)

%o (PARI) a(n)=if(n<1,0,numerator(bernfrac(2*n)/(2*n)/(2*n-1)))

%o (Magma) [Numerator(Bernoulli(2*n)/(2*n*(2*n-1))): n in [1..25]]; // _G. C. Greubel_, Sep 19 2019

%o (Sage) [numerator(bernoulli(2*n)/(2*n*(2*n-1))) for n in (1..25)] # _G. C. Greubel_, Sep 19 2019

%o (GAP) List([1..25], n-> NumeratorRat(Bernoulli(2*n)/(2*n*(2*n-1))) ); # _G. C. Greubel_, Sep 19 2019

%Y Denominators given by A046969.

%Y Similar to but different from A001067. See A090495, A090496.

%K frac,sign,nice

%O 1,6

%A Douglas Stoll (dougstoll(AT)email.msn.com)

%E More terms from _Frank Ellermann_, Jun 13 2001