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%I #10 Aug 30 2019 03:31:18
%S 1,-1,2,-3,3360,-995040,39916800,-656924748480,1214047650816000,
%T -169382556838010880,15749593891765493760000,
%U -4054844479616799289344000,34017686450062663131463680000,-11402327189708082115897599590400000,189528830020089532044244068728832000000
%N Numerators of rationals used in the Euler-Maclaurin type derivation of Stirling's formula for N!.
%C Denominators are given in A097302.
%C The e.g.f. sum( A(2*n+1)*(x^(2*n+1))/(2*n+1)!,n=0..infinity) appears in the Stirling-formula derivation for N! with x=1/N in the exponent and the formula for A(2*n+1):=a(n)/A097302(n), n>=0, is given below. For Stirling's formula see A001163 and A001164.
%C The rationals A(2*n+1) = B(n):= (2*n)!*Bernoulli(2*(n+1))/(2*(n+1)) = a(n)/A097304(n) with A(2*n):=0 are the logarithmic transform of the rational sequence {A001163(n)/A001164(n)} (inverse of the sequence transform EXP)
%D Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, p. 87.
%H W. Lang, <a href="/A097301\a097301.txt">More terms and comments</a>.
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F a(n)=numerator(B(n)) with B(n):=Bernoulli(2*n+2)*(2*n)!/(2*n+2) and Bernoulli(n)= A027641(n)/A027642(n).
%K sign,frac,easy
%O 0,3
%A _Wolfdieter Lang_, Aug 13 2004