A307974 Inverse binomial transform of the "original" Bernoulli numbers [A164555(n)/A027642(n)] with 1 and 1/2 swapped. Numerators.
1, 1, -4, 2, -38, 3, -73, 4, -68, 5, -179, 6, -9218, 7, -19, 8, -3976, 9, 18143, 10, -89038, 11, 426463, 12, -118199108, 13, 4276511, 14, -11874736822, 15, 4307920527007, 16, -3854660524816, 17, 1288843929131, 18, -13157635776544491194, 19, 1464996956920721, 20, -130541359248224699708
Offset: 0
Examples
Successive differences show the data in the first column:
1/2, 1, 1/6, 0, -1/30, 0, 1/42, 0, ...
1/2, -5/6, -1/6, -1/30, 1/30, 1/42, ...
-4/3, 2/3, 2/15, 1/15, -1/105, ...
2, -8/15, -1/15, -8/105, ...
-38/15, 7/15, -1/105, ...
3, -10/21, ...
-73/21, ...
... .
The third column is A256671(n)/A256675(n).
Crossrefs
Programs
-
Mathematica
m = 40; b[n_] = BernoulliB[n]; b[0] = 1/2; b[1] = 1; a[n_] := Sum[(-1)^(n - k)*Binomial[n, k]*b[k], {k, 0, m}] // Numerator; Table[a[n], {n, 0, m}] (* Second program: *) m = 40; bb = CoefficientList[Series[x/(1 - Exp[-x]), {x, 0, m}], x]*Range[0, m]!; bb[[1]] = 1/2; bb[[2]] = 1; a[n_] := Differences[bb, n][[1]] // Numerator; Table[a[n], {n, 0, m}] (* Jean-François Alcover, May 31 2019 *)
Formula
a(2*n+1) = n+1 (conjectured).
Comments