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 A141591 New triangular sequence of coefficients based on A123125 Eulerian numbers as: ( Like A109128 to the Binomials) t(n,m)=2*A123125(n,m)-1. 0
 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, 2, -1, -1, 2, 114, 604, 604, 114, 2, -1, -1, 2, 240, 2382, 4832, 2382, 240, 2, -1, -1, 2, 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row sums are: {1, 0, 0, 2, 10, 46, 238, 1438, 10078, 80638, 725758, 7257598}; One coefficient in m and one in n are added to make a complete symmetrical triangle of coefficients. REFERENCES Douglas C. Montgomery, Lynwood A, Johnson, Forecasting and Time Series Analysis,McGraw-Hill, New York,1976,page 91 LINKS FORMULA t(n,m)=2*A123125(n,m)-1. EXAMPLE {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, 2, -1}, {-1, 2, 114, 604, 604, 114, 2, -1}, {-1, 2, 240, 2382, 4832, 2382, 240, 2, -1}, {-1, 2, 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, 2620708, 910384, 95680, 2026, 2, -1} MATHEMATICA 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] CROSSREFS Cf. A109128. Sequence in context: A214248 A152719 A107044 * A174545 A102523 A323023 Adjacent sequences:  A141588 A141589 A141590 * A141592 A141593 A141594 KEYWORD tabl,uned,sign AUTHOR Roger L. Bagula and Gary W. Adamson, Aug 20 2008 STATUS approved

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Last modified September 22 12:59 EDT 2021. Contains 347607 sequences. (Running on oeis4.)