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A144395
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A designed polynomial set that gives a {1,6,1} quadratic and gives a symmetrical triangle of coefficients: p(x,n)=If[n == 2, 1, ((x + 1)^n -If[n == 0, 1, x^n + (n - 1)*x^(n - 1) + (n - 1)*x + 1])/x],.
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0
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1, 1, 1, 1, 6, 1, 1, 10, 10, 1, 1, 15, 20, 15, 1, 1, 21, 35, 35, 21, 1, 1, 28, 56, 70, 56, 28, 1, 1, 36, 84, 126, 126, 84, 36, 1, 1, 45, 120, 210, 252, 210, 120, 45, 1, 1, 55, 165, 330, 462, 462, 330, 165, 55, 1, 1, 66, 220, 495, 792, 924, 792, 495, 220, 66, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Row sums are:{1, 2, 8, 22, 52, 114, 240, 494, 1004, 2026, 4072}.
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FORMULA
| p(x,n)=If[n == 2, 1, ((x + 1)^n -If[n == 0, 1, x^n + (n - 1)*x^(n - 1) + (n - 1)*x + 1])/x]; t(n,m)=coefficients(p(x,n)).
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EXAMPLE
| {1},
{1, 1},
{1, 6, 1},
{1, 10, 10, 1},
{1, 15, 20, 15, 1},
{1, 21, 35, 35, 21, 1},
{1, 28, 56, 70, 56, 28, 1},
{1, 36, 84, 126, 126, 84, 36, 1},
{1, 45, 120, 210, 252, 210, 120, 45, 1},
{1, 55, 165, 330, 462, 462, 330, 165, 55, 1},
{1, 66, 220, 495, 792, 924, 792, 495, 220, 66, 1}
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MATHEMATICA
| Clear[p, x, n] p[x_, n_] = If[ n == 2, 1, ((x + 1)^n - If[n == 0, 1, x^n + (n - 1)*x^(n - 1) + (n - 1)*x + 1])/x]; Table[ExpandAll[p[x, n]], {n, 2, 12}]; Table[CoefficientList[p[x, n], x], {n, 2, 12}]; Flatten[%]
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CROSSREFS
| Sequence in context: A174377 A176151 A204001 * A046621 A046617 A131063
Adjacent sequences: A144392 A144393 A144394 * A144396 A144397 A144398
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KEYWORD
| nonn,uned
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AUTHOR
| Roger L. Bagula and Gary W. Adamson (rlbagulatftn(AT)yahoo.com), Oct 02 2008
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