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Search: id:A141591
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| A141591 |
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New triangular sequence of coefficients based on A123125 Eulerian numbers as: ( Like A109128 to the Binomials) t(n,m)=2*A123125(n,m)-1. |
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1
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| 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; listen)
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OFFSET
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1,5
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COMMENT
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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.
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REFERENCES
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Douglas C. Montgomery, Lynwood A, Johnson, Forecasting and Time Series Analysis,McGraw-Hill, New York,1976,page 91
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FORMULA
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t(n,m)=2*A123125(n,m)-1.
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EXAMPLE
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{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}
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A109128.
Sequence in context: A103444 A099172 A107044 this_sequence A102523 A083415 A115514
Adjacent sequences: A141588 A141589 A141590 this_sequence A141592 A141593 A141594
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KEYWORD
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tabl,uned,sign
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AUTHOR
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Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 20 2008
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