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%I #2 Oct 12 2012 14:54:51
%S 1,1,-1,-1,2,-1,-1,2,2,-1,-1,2,8,2,-1,-1,2,22,22,2,-1,-1,2,52,132,52,
%T 2,-1,-1,2,114,604,604,114,2,-1,-1,2,240,2382,4832,2382,240,2,-1,-1,2,
%U 494,8586,31238,31238,8586,494,2,-1,-1,2,1004,29216,176468,312380,176468,29216,1004,2,-1,-1,2,2026,95680,910384,2620708
%N New triangular sequence of coefficients based on A123125 Eulerian numbers as: ( Like A109128 to the Binomials) t(n,m)=2*A123125(n,m)-1.
%C Row sums are:
%C {1, 0, 0, 2, 10, 46, 238, 1438, 10078, 80638, 725758, 7257598};
%C One coefficient in m and one in n are added to make a complete symmetrical
%C triangle of coefficients.
%D Douglas C. Montgomery, Lynwood A, Johnson, Forecasting and Time Series Analysis,McGraw-Hill, New York,1976,page 91
%F t(n,m)=2*A123125(n,m)-1.
%e {1},
%e {1, -1},
%e {-1,2, -1},
%e {-1, 2, 2, -1},
%e {-1, 2, 8, 2, -1},
%e {-1, 2, 22, 22, 2, -1},
%e {-1, 2, 52, 132, 52, 2, -1},
%e {-1, 2, 114, 604, 604, 114, 2, -1},
%e {-1, 2, 240, 2382, 4832, 2382, 240, 2, -1},
%e {-1, 2, 494, 8586, 31238,31238, 8586, 494, 2, -1},
%e {-1, 2, 1004, 29216, 176468, 312380, 176468, 29216, 1004, 2, -1},
%e {-1, 2, 2026, 95680, 910384, 2620708, 2620708, 910384, 95680, 2026, 2, -1}
%t Clear[f, x, n, a] f[x_, n_] := f[x, n] = (1 - x)^(n + 1)*Sum[k^n*x^k, {k, 0, Infinity}]; Table[FullSimplify[ExpandAll[f[x, n]]], {n, 0, 10}]; a = Join[{{1}}, Table[Join[CoefficientList[FullSimplify[2*ExpandAll[f[x, n]]] - 1, x], {-1}], {n, 0, 10}]]; Flatten[a]
%Y Cf. A109128.
%K tabl,uned,sign
%O 1,5
%A _Roger L. Bagula_ and _Gary W. Adamson_, Aug 20 2008
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