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

0,3

Comments

Denominators: 2, 2, 3, 1, 15, 1, 21, 1, 15, 1, 33, 1, ... .

Denominators 3, 15, 21, 15, 33, 1365, 3, 255, ... coincide with cosecant numbers A001897, except 1 (conjectured).

Links

Table of n, a(n) for n=0..40.

Formula

a(2*n+1) = n+1 (conjectured).

Example

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).

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 *)

Crossrefs

Cf. A001897, A027642, A164555, A176328 (for the second bisection), A256671/A256675, A306821 (denominators).

Sequence in context: A364878 A102015 A302890 * A303625 A123850 A280780

Adjacent sequences: A307971 A307972 A307973 * A307975 A307976 A307977

Keyword

sign,frac

Author

Paul Curtz, May 30 2019

Status

approved