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A229054 Autosequence preceding -A226158(n).

%I A229054 #17 Oct 05 2013 04:45:48
%S A229054 0,0,0,1,2,2,1,1,4,4,-13,-13,142,142,-1931,-1931,36296,36296,-893273,
%T A229054 -893273,27927346,27927346,-1081725559,-1081725559,50861556172,
%U A229054 50861556172,-2854289486309,-2854289486309
%N A229054 Autosequence preceding -A226158(n).
%C A229054 Extension of the difference table of Genocchi numbers A226158(n). The signs are changed.
%C A229054 Consider the difference table of -A226158:
%C A229054 0,   1,  1,  0  -1,  0,
%C A229054 1,   0, -1, -1,  1,  3,
%C A229054 -1, -1,  0,  2,  2, -6,
%C A229054 0,   1,  2,  0, -8, -8,
%C A229054 1,   1, -2, -8,  0, 56,
%C A229054 0,  -3, -6,  8, 56,  0, etc.
%C A229054 Upon the table, we prolonged the main diagonal by 0 followed by 0 on the same row. Hence
%C A229054 0,  0,  0,  1,  2,   2,    1,    1,
%C A229054 0,  0,  1,  1,  0,  -1,    0,    3,    = 0 followed by -A226158.
%C A229054 0,  1,  0, -1, -1,   1,    3,   -3,
%C A229054 1, -1, -1,  0,  2,   2,   -6,  -14,
%C A229054 -2, 0,  1,  2,  0,  -8,   -8,   48,
%C A229054 2,  1,  1, -2, -8,   0,   56,   56,
%C A229054 -1, 0, -3, -6,  8,  56,    0, -608,
%C A229054 1, -3, -3, 14, 48, -56, -608,    0,  etc.
%C A229054 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.
%F A229054 a(n) = 0, 0, 0 followed by (-1)^n * A133135(n).
%t A229054 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}]]
%t A229054 (* or *)
%t A229054 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 *)
%K A229054 sign
%O A229054 0,5
%A A229054 _Paul Curtz_, Sep 12 2013