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A307974 Inverse binomial transform of the "original" Bernoulli numbers [A164555(n)/A027642(n)] with 1 and 1/2 swapped. Numerators.

%I A307974 #37 Jun 04 2019 05:59:15
%S A307974 1,1,-4,2,-38,3,-73,4,-68,5,-179,6,-9218,7,-19,8,-3976,9,18143,10,
%T A307974 -89038,11,426463,12,-118199108,13,4276511,14,-11874736822,15,
%U A307974 4307920527007,16,-3854660524816,17,1288843929131,18,-13157635776544491194,19,1464996956920721,20,-130541359248224699708
%N A307974 Inverse binomial transform of the "original" Bernoulli numbers [A164555(n)/A027642(n)] with 1 and 1/2 swapped. Numerators.
%C A307974 Denominators: 2, 2, 3, 1, 15, 1, 21, 1, 15, 1, 33, 1, ... .
%C A307974 Denominators 3, 15, 21, 15, 33, 1365, 3, 255, ... coincide with cosecant numbers A001897, except 1 (conjectured).
%F A307974 a(2*n+1) = n+1 (conjectured).
%e A307974 Successive differences show the data in the first column:
%e A307974     1/2,     1,    1/6,     0,    -1/30,   0,   1/42, 0, ...
%e A307974     1/2,   -5/6,  -1/6,   -1/30,   1/30,  1/42, ...
%e A307974    -4/3,    2/3,   2/15,   1/15,  -1/105, ...
%e A307974      2,    -8/15, -1/15,  -8/105, ...
%e A307974   -38/15,   7/15, -1/105, ...
%e A307974      3,   -10/21, ...
%e A307974   -73/21, ...
%e A307974   ... .
%e A307974 The third column is A256671(n)/A256675(n).
%t A307974 m = 40;
%t A307974 b[n_] = BernoulliB[n]; b[0] = 1/2; b[1] = 1;
%t A307974 a[n_] := Sum[(-1)^(n - k)*Binomial[n, k]*b[k], {k, 0, m}] // Numerator;
%t A307974 Table[a[n], {n, 0, m}]
%t A307974 (* Second program: *)
%t A307974 m = 40;
%t A307974 bb = CoefficientList[Series[x/(1 - Exp[-x]), {x, 0, m}], x]*Range[0, m]!;
%t A307974 bb[[1]] = 1/2; bb[[2]] = 1;
%t A307974 a[n_] := Differences[bb, n][[1]] // Numerator;
%t A307974 Table[a[n], {n, 0, m}] (* _Jean-François Alcover_, May 31 2019 *)
%Y A307974 Cf. A001897, A027642, A164555, A176328 (for the second bisection), A256671/A256675, A306821 (denominators).
%K A307974 sign,frac
%O A307974 0,3
%A A307974 _Paul Curtz_, May 30 2019