A229054 Autosequence preceding -A226158(n).

0, 0, 0, 1, 2, 2, 1, 1, 4, 4, -13, -13, 142, 142, -1931, -1931, 36296, 36296, -893273, -893273, 27927346, 27927346, -1081725559, -1081725559, 50861556172, 50861556172, -2854289486309, -2854289486309

Offset: 0

Author

Paul Curtz, Sep 12 2013

Keywords

• sign

Comments

Extension of the difference table of Genocchi numbers A226158(n). The signs are changed.

Consider the difference table of -A226158:

0, 1, 1, 0 -1, 0,

1, 0, -1, -1, 1, 3,

-1, -1, 0, 2, 2, -6,

0, 1, 2, 0, -8, -8,

1, 1, -2, -8, 0, 56,

0, -3, -6, 8, 56, 0, etc.

Upon the table, we prolonged the main diagonal by 0 followed by 0 on the same row. Hence

0, 0, 0, 1, 2, 2, 1, 1,

0, 0, 1, 1, 0, -1, 0, 3, = 0 followed by -A226158.

0, 1, 0, -1, -1, 1, 3, -3,

1, -1, -1, 0, 2, 2, -6, -14,

-2, 0, 1, 2, 0, -8, -8, 48,

2, 1, 1, -2, -8, 0, 56, 56,

-1, 0, -3, -6, 8, 56, 0, -608,

1, -3, -3, 14, 48, -56, -608, 0, etc.

The first row, a(n), is equal to its inverse binomial transform signed, the main diagonal of the difference table is composed of 0's, so it is an autosequence of the first kind.

Programs

Mathematica

max = 24; p[0, ] = 1; p[n, x_] := (1+x)*((1+x)^(n-1) + x^(n-1))/2; t = Table[Coefficient[p[n, x], x, k], {n, 0, max+2}, {k, 0, max+2}]; a[n_] := Inverse[t][[All, 3]][[n+3]]; A133135 = Table[a[n], {n, 0, max}]; Join[{0, 0, 0}, Table[(-1)^n*A133135[[n+1]], {n, 0, max}]]
(* or *)
g[n_ /; n < 3] = 0; g[3] = -1; g[n_] := (n-2)*EulerE[n-3, 0]; Table[-g[n], {n, 0, 27}] // Accumulate (* Jean-François Alcover, Sep 12 2013 *)

Formula

a(n) = 0, 0, 0 followed by (-1)^n * A133135(n).