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%I #21 Feb 22 2019 02:01:05
%S -1,-1,1,1,59,3,169,5,179,7,533,9,26609,11,79,13,3523,15,56635,17,
%T -168671,19,857273,21,-236304031,23,8553247,25,-23749438409,27,
%U 8615841677021,29,-7709321025917,31,2577687858559,33,-26315271552988224913
%N Let b(k) = A164555(k)/A027642(k), the sequence of "original" Bernoulli numbers with -1 instead of A164555(0)=1; then a(n) = numerator of the n-th term of the binomial transform of the b(k) sequence.
%C (a(n)/A027642(n)) = -1, -1/2, 1/6, 1, 59/30, 3, 169/42, 5, 179/30, 7, 533/66, 9,.. .
%C Difference table for a(n)/A027642(n):
%C -1, -1/2, 1/6, 1, 59/30, 3, 169/42, ...
%C 1/2, 2/3, 5/6, 29/30, 31/30, 43/42, 41/42, ... = A165161(n)/A051717(n+1)
%C 1/6, 1/6, 2/15, 1/15, -1/105, -1/21, -1/105, ... not in the OEIS
%C 0, -1/30, -1/15, -8/105, -4/105, 4/105, 8/105, ... etc.
%C Compare with the array in A190339.
%F (a(n+1) - a(n))/A027642(n) = A165161(n)/A051717(n+1).
%F (A164558(n) - a(n))/A027642(n) = 2's = A007395.
%F (a(n) - A164555(n))/A027642(n) = n - 2 = A023444(n).
%t b[0] = -1; b[1] = 1/2; b[n_] := BernoulliB[n]; a[n_] := Sum[Binomial[n, k]*b[k], {k, 0, n}] // Numerator; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jan 30 2014 *)
%Y Cf. A164558/A027642, A165161(n)/A051717(n+1).
%K sign,frac
%O 0,5
%A _Paul Curtz_, Jan 15 2014
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