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A158208
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A symmetrical triangle of polynomial coefficients: p(x,n)=If[n == 0, 2, Sum[Binomial[n, i]*(x - 1)^i, {i, 0, Floor[(n - 1)/2]}] + x^n*Sum[ Binomial[n, i]*(1/x - 1)^i, {i, 0, Floor[(n - 1)/2]}]].
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0
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2, 1, 1, 1, 0, 1, -2, 3, 3, -2, -3, 4, 0, 4, -3, 6, -15, 10, 10, -15, 6, 10, -24, 15, 0, 15, -24, 10, -20, 70, -84, 35, 35, -84, 70, -20, -35, 120, -140, 56, 0, 56, -140, 120, -35, 70, -315, 540, -420, 126, 126, -420, 540, -315, 70, 126, -560, 945, -720, 210, 0, 210
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,1
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COMMENTS
| Straight row sums are two, but absolute value row sums are:
{2, 2, 2, 10, 14, 62, 98, 418, 702, 2942, 5122,...}.
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FORMULA
| p(x,n)=If[n == 0, 2, Sum[Binomial[n, i]*(x - 1)^i, {i, 0, Floor[(n - 1)/2]}] + x^n*Sum[ Binomial[n, i]*(1/x - 1)^i, {i, 0, Floor[(n - 1)/2]}]];
out(n,m)=coefficients(p(x,n)).
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EXAMPLE
| {2},
{1, 1},
{1, 0, 1},
{-2, 3, 3, -2},
{-3, 4, 0, 4, -3},
{6, -15, 10, 10, -15, 6},
{10, -24, 15, 0, 15, -24, 10},
{-20, 70, -84, 35, 35, -84, 70, -20},
{-35, 120, -140, 56, 0, 56, -140, 120, -35},
{70, -315, 540, -420, 126, 126, -420, 540, -315, 70},
{126, -560, 945, -720, 210, 0, 210, -720, 945, -560, 126}
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MATHEMATICA
| Clear[p, x, n];
p[x_, n_] = If[ n == 0, 2, Sum[Binomial[ n, i]*(x - 1)^i, {i, 0, Floor[(n - 1)/2]}] + Expand[x^n*Sum[Binomial[n, i]*(1/x - 1)^ i, {i, 0, Floor[(n - 1)/2]}]]];
Table[CoefficientList[p[x, n], x], {n, 0, 10}];
Flatten[%]
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CROSSREFS
| Sequence in context: A093718 A035212 A068029 * A117274 A140883 A064744
Adjacent sequences: A158205 A158206 A158207 * A158209 A158210 A158211
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KEYWORD
| sign,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 13 2009
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