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A155865
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A sequence of polynomial coefficients related to the first derivative of the Pascal triangle: p(x,n)=x^n+1+x*d(x+1)^(n+1)/dx=If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n - 1), {x, 1}]].
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0
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1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 3, 6, 3, 1, 1, 4, 12, 12, 4, 1, 1, 5, 20, 30, 20, 5, 1, 1, 6, 30, 60, 60, 30, 6, 1, 1, 7, 42, 105, 140, 105, 42, 7, 1, 1, 8, 56, 168, 280, 280, 168, 56, 8, 1, 1, 9, 72, 252, 504, 630, 504, 252, 72, 9, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,8
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COMMENTS
| Row sums are:
{1, 2, 3, 6, 14, 34, 82, 194, 450, 1026, 2306,...}
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FORMULA
| p(x,n)=x^n+1+x*d(x+1)^(n)/dx
p(x,n)=If[n == 0, 1, x^n + 1 + x*D[(x + 1)^(n - 1), {x, 1}]]
t(n,m)=coefficients(p(x,n))
Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 09 2010: (Start)
c(n)=If[n == 0, 1, If[n == 1, 1, Product[(i - 1), {i, 2, n}]]];
t(n,m)=c(n)/(c(m)*c(n-m)) (End)
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EXAMPLE
| {1},
{1, 1},
{1, 1, 1},
{1, 2, 2, 1},
{1, 3, 6, 3, 1},
{1, 4, 12, 12, 4, 1},
{1, 5, 20, 30, 20, 5, 1},
{1, 6, 30, 60, 60, 30, 6, 1},
{1, 7, 42, 105, 140, 105, 42, 7, 1},
{1, 8, 56, 168, 280, 280, 168, 56, 8, 1},
{1, 9, 72, 252, 504, 630, 504, 252, 72, 9, 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, 1}]];
Table[ExpandAll[p[x, n]], {n, 0, 10}];
a = Table[CoefficientList[ExpandAll[p[x, n]], x], {n, 0, 10}];
Flatten[a]
Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Mar 09 2010: (Start)
(* at q=1*)
c[n_, q_] = If[n == 0, 1, If[n == 1, 1, Product[(i - 1)^q, {i, 2, n}]]];
t[n_, m_, q_] = c[n, q]/(c[m, q]*c[n - m, q]);
Table[Flatten[Table[Table[t[n, m, q], {m, 0, n}], {n, 0, 10}]], {q, 1, 10}] (End)
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CROSSREFS
| Sequence in context: A008302 A131791 A010358 * A156133 A010048 A055870
Adjacent sequences: A155862 A155863 A155864 * A155866 A155867 A155868
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KEYWORD
| nonn,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 29 2009
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