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A155467
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Triangle read by rows: f(n)=(n+1)!;a(n,m)=f(n)/(f(m)*f(n-m)); e(n,m)=Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; t(n,m)=e(n,m)*a(n,m).
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1, 1, 1, 1, 6, 1, 1, 22, 22, 1, 1, 65, 220, 65, 1, 1, 171, 1510, 1510, 171, 1, 1, 420, 8337, 21140, 8337, 420, 1, 1, 988, 40068, 218666, 218666, 40068, 988, 1, 1, 2259, 175296, 1852914, 3935988, 1852914, 175296, 2259, 1, 1, 5065, 717600, 13655760, 55034868
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Row sums are:A099765;
{1, 2, 8, 46, 352, 3364, 38656, 519446, 7996928, 138826588, 2683604992,...}.
The sequence substitutes Eulerian numbers for the binomial in a triangle of
Narayana numbers A001263,
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FORMULA
| f(n)=(n+1)!;a(n,m)=f(n)/(f(m)*f(n-m));
e(n,m)=Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
t(n,m)=e(n,m)*a(n,m).
t(n,m)=Eulerian[n + 1, m]*Binomial[n + 1, m]/(m + 1) [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 14 2010]
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EXAMPLE
| {1},
{1, 1},
{1, 6, 1},
{1, 22, 22, 1},
{1, 65, 220, 65, 1},
{1, 171, 1510, 1510, 171, 1},
{1, 420, 8337, 21140, 8337, 420, 1},
{1, 988, 40068, 218666, 218666, 40068, 988, 1},
{1, 2259, 175296, 1852914, 3935988, 1852914, 175296, 2259, 1},
{1, 5065, 717600, 13655760, 55034868, 55034868, 13655760, 717600, 5065, 1},
{1, 11198, 2798345, 90893880, 642715524, 1210767096, 642715524, 90893880, 2798345, 11198, 1}
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MATHEMATICA
| t[n_, k_] = Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}];
f[n_] = Product[k + 1, {k, 0, n}];
A[n_, m_] = f[n]/(f[m]*f[n - m]);
Table[Table[t[n + 1, m]*A[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]
Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Apr 14 2010: (Start)
<< DiscreteMath`Combinatorica`
t[n_, m_] := Eulerian[n + 1, m]*Binomial[n + 1, m]/(m + 1);
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%] (End)
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CROSSREFS
| A001263, A099765
Sequence in context: A157638 A142596 A176063 * A152936 A152969 A138076
Adjacent sequences: A155464 A155465 A155466 * A155468 A155469 A155470
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KEYWORD
| nonn,tabl,uned
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AUTHOR
| Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Jan 22 2009
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