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A141581
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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}].
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0
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Row sums are are zero.
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REFERENCES
| Douglas C. Montgomery, Lynwood A, Johnson, Forecasting and Time Series Analysis,McGraw-Hill, New York,1976,page 91
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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).
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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}
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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]
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CROSSREFS
| Cf. A123125.
Sequence in context: A083056 A061896 A069850 * A179286 A193690 A108964
Adjacent sequences: A141578 A141579 A141580 * A141582 A141583 A141584
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KEYWORD
| tabf,sign
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Aug 19 2008
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