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A155863
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A sequence of polynomial coefficients related to the third derivative of the Pascal triangle: p(x,n)=x^n+1+x*d^3(x+1)^(n+1)/dx^3=If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n + 1), {x, 3}]].
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1
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1, 1, 1, 1, 6, 1, 1, 24, 24, 1, 1, 60, 120, 60, 1, 1, 120, 360, 360, 120, 1, 1, 210, 840, 1260, 840, 210, 1, 1, 336, 1680, 3360, 3360, 1680, 336, 1, 1, 504, 3024, 7560, 10080, 7560, 3024, 504, 1, 1, 720, 5040, 15120, 25200, 25200, 15120, 5040, 720, 1, 1, 990, 7920
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are:
{1, 2, 8, 50, 242, 962, 3362, 10754, 32258, 92162, 253442,...}
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FORMULA
| p(x,n)=x^n+1+x*d^3(x+1)^(n+1)/dx^3
p(x,n)=If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n + 1), {x, 3}]]
t(n,m)=coefficients(p(x,n))
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EXAMPLE
| {1},
{1, 1},
{1, 6, 1},
{1, 24, 24, 1},
{1, 60, 120, 60, 1},
{1, 120, 360, 360, 120, 1},
{1, 210, 840, 1260, 840, 210, 1},
{1, 336, 1680, 3360, 3360, 1680, 336, 1},
{1, 504, 3024, 7560, 10080, 7560, 3024, 504, 1},
{1, 720, 5040, 15120, 25200, 25200, 15120, 5040, 720, 1},
{1, 990, 7920, 27720, 55440, 69300, 55440, 27720, 7920, 990, 1}
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MATHEMATICA
| Clear[p, n, m, x, a];
p[x_, n_] = If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n + 1), {x, 3}]];
Table[ExpandAll[p[x, n]], {n, 0, 10}];
a = Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}];
Flatten[a]
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CROSSREFS
| Sequence in context: A060187 A174527 A156139 * A173882 A174045 A169660
Adjacent sequences: A155860 A155861 A155862 * A155864 A155865 A155866
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KEYWORD
| nonn,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 29 2009
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