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A156901
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A infinite sum polynomial triangle of coefficients: p(x,n)=p[x_, n_] = ((1 + x - x^3)^ (n + 1))*Sum[(k + 1)^n*(-x + x^3)^k, {k, 0, Infinity}].
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0
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1, 1, 1, -2, 1, 1, -8, 8, -4, 1, 1, -22, 55, -52, 23, -6, 1, 1, -52, 290, -472, 394, -188, 50, -8, 1, 1, -114, 1265, -3624, 4838, -3668, 1750, -536, 97, -10, 1, 1, -240, 4884, -24092, 49239, -56448, 40664, -19320, 6231, -1360, 180, -12, 1, 1, -494, 17419
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Row sums are:
{1, 1, 0, -2, 0, 16, 0, -272, 0, 7936, 0,...}.
Second column is negative second order Eulerian numbers;A005803:
{2, 8, 22, 52, 114, 240, 494, 1004, 2026,...}.
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FORMULA
| p(x,n)=((1 + 2*x - x^2)^ (n + 1))*Sum[(k + 1)^n*(-2*x + x^2)^k, {k, 0, Infinity}];
t*n,m)=coefficients(p(x,n)).
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EXAMPLE
| {1},
{1},
{1, -2, 1},
{1, -8, 8, -4, 1},
{1, -22, 55, -52, 23, -6, 1},
{1, -52, 290, -472, 394, -188, 50, -8, 1},
{1, -114, 1265, -3624, 4838, -3668, 1750, -536, 97, -10, 1},
{1, -240, 4884, -24092, 49239, -56448, 40664, -19320, 6231, -1360, 180, -12, 1},
{1, -494, 17419, -142124, 441625, -730898, 749723, -515944, 247067, -83122, 19673, -3244, 331, -14, 1},
{1, -1004, 58934, -764304, 3572456, -8350972, 11830426, -11177232, 7416622, -3557988, 1247626, -317648, 57896, -7476, 614, -16, 1},
{1, -2026, 192373, -3832896, 26475808, -86593808, 165864296, -208831536, 184662242, -119120564, 57229874, -20677408, 5613688, -1127616, 163312, -16880, 1157, -18, 1}
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MATHEMATICA
| Clear[p, x, n, m];
p[x_, n_] = ((1 + 2*x - x^2)^ (n + 1))*Sum[(k + 1)^n*(-2*x + x^2)^k, {k, 0, Infinity}];
Table[Expand[FullSimplify[ExpandAll[p[x, n]]]], {n, 0, 10}];
Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
Flatten[%]
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CROSSREFS
| Sequence in context: A051428 A176698 A129276 * A167400 A165889 A087127
Adjacent sequences: A156898 A156899 A156900 * A156902 A156903 A156904
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KEYWORD
| sign,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Feb 17 2009
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