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.

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

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

Table of n, a(n) for n=1..72.

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