A141581 Cosine projection of Eulerian numbers A123125 as coefficient triangle: w=1: f(x,n)=(1 - 2*Cos[w]*x + x^2)^(n + 1)*Sum[k^n*x^k*Cos[w], {k, 0, Infinity}].

1, -1, 0, 1, -2, 1, 0, 1, -2, 0, 2, -1, 0, 1, 0, -9, 16, -9, 0, 1, 0, 1, 6, -34, 46, 0, -46, 34, -6, -1, 0, 1, 20, -75, 0, 330, -552, 330, 0, -75, 20, 1, 0, 1, 50, -76, -650, 2325, -2652, 0, 2652, -2325, 650, 76, -50, -1, 0, 1, 112, 259, -3808, 8561, -112, -26229, 42432, -26229, -112, 8561, -3808, 259, 112, 1, 0, 1, 238, 2106

Offset

1,5

Comments

Row sums are zero.

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..76.

Formula

w=1: f(x,n)=(1 - 2*Cos[w]*x + x^2)^(n + 1)*Sum[k^n*x^k*Cos[w], {k, 0, Infinity}]; t(n,m)=Coefficients(f(x,n).

Example

{1, -1},
{0, 1, -2, 1},
{0, 1, -2, 0, 2, -1},
{0, 1, 0, -9, 16, -9, 0, 1},
{0, 1, 6, -34, 46, 0, -46, 34, -6, -1},
{0, 1, 20, -75, 0, 330, -552, 330, 0, -75, 20, 1},
{0, 1, 50, -76, -650, 2325, -2652, 0, 2652, -2325, 650, 76, -50, -1},
{0, 1, 112, 259, -3808, 8561, -112, -26229, 42432, -26229, -112, 8561, -3808, 259, 112, 1},
{0, 1, 238, 2106, -14210, 8974, 96390, -278222, 288118, 0, -288118, 278222, -96390, -8974,14210, -2106, -238, -1},
{0, 1,492, 9633, -35376, -128820, 849072, -1485876, 35376, 3636678, -5762360, 3636678, 35376, -1485876, 849072, -128820, -35376, 9633, 492, 1},
{0, 1, 1002, 36752, -15498, -1232373, 4372248, -1676256, -20519256, 52187994, -50774756, 0, 50774756, -52187994, 20519256, 1676256, -4372248, 1232373, 15498, -36752, -1002, -1}

Mathematica

Clear[f, x, n, a] w = 0; f[x_, n_] := f[x, n] = (1 - 2*Cos[w]*x + x^2)^(n + 1)*Sum[k^n*x^k*Cos[w], {k, 0, Infinity}]; Table[FullSimplify[ExpandAll[f[x, n]]], {n, 0, 10}]; a = Table[CoefficientList[FullSimplify[ExpandAll[f[x, n]]], x], {n, 0, 10}]; Flatten[a]

Crossrefs

Cf. A123125.

Sequence in context: A061896 A366793 A069850 * A179286 A193690 A108964

Adjacent sequences: A141578 A141579 A141580 * A141582 A141583 A141584

Keyword

tabf,sign

Author

Roger L. Bagula and Gary W. Adamson, Aug 19 2008

Status

approved