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A176327 Numerators of the rational sequence with e.g.f. (x/2)*(1+exp(-x))/(1-exp(-x)).

%I A176327 #26 Oct 21 2017 21:15:10
%S A176327 1,0,1,0,-1,0,1,0,-1,0,5,0,-691,0,7,0,-3617,0,43867,0,-174611,0,
%T A176327 854513,0,-236364091,0,8553103,0,-23749461029,0,8615841276005,0,
%U A176327 -7709321041217,0,2577687858367,0,-26315271553053477373,0,2929993913841559,0,-261082718496449122051
%N A176327 Numerators of the rational sequence with e.g.f. (x/2)*(1+exp(-x))/(1-exp(-x)).
%C A176327 Numerator of the Bernoulli number B_n, except B(1)=0.
%C A176327 A027641 is the main entry for this sequence, which is only a minor variation. - _N. J. A. Sloane_, Nov 29 2010.
%C A176327 This could formally be defined by building the arithmetic mean of the numerators in A164555(n) and A027641(n).
%H A176327 Antti Karttunen, <a href="/A176327/b176327.txt">Table of n, a(n) for n = 0..200</a>
%F A176327 a(2n+1)  = 0. a(2n ) = A000367(n).
%F A176327 a(n) = A164555(n) = A027641(n) if n <>1.
%p A176327 seq(numer((bernoulli(i,0)+bernoulli(i,1))/2),i=0..40); # _Peter Luschny_, Jun 17 2012
%t A176327 terms = 41; egf = (x/2)*((1 + Exp[-x])/(1 - Exp[-x])) + O[x]^(terms+1);
%t A176327 CoefficientList[egf, x]*Range[0, terms-1]! // Numerator (* _Jean-François Alcover_, Jun 13 2017 *)
%o A176327 (PARI) apply(numerator, Vec(serlaplace((x/2)*(1+exp(-x))/(1-exp(-x))))) \\ _Charles R Greathouse IV_, Sep 26 2017
%Y A176327 Cf. A176289 (denominators), A027642, A141056, A164020, A165823
%K A176327 sign
%O A176327 0,11
%A A176327 _Paul Curtz_, Apr 15 2010
%E A176327 New name from _Peter Luschny_, Jun 18 2012