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

0,5

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.

Links

Table of n, a(n) for n=0..27.

Formula

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

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 *)

Crossrefs

Sequence in context: A293208 A247364 A301895 * A133135 A292189 A284992

Adjacent sequences: A229051 A229052 A229053 * A229055 A229056 A229057

Keyword

sign

Author

Paul Curtz, Sep 12 2013

Status

approved