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A090238
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Triangle T(n,k), 0<=k<=n, read by rows, given by [0, 2, 1, 3, 2, 4, 3, 5, 4, 6, 5, 7, 6, ...] DELTA [1, 0, 0, 0, 0, 0, 0, 0, 0, ...] where DELTA is the operator defined in A084938.
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11
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1, 0, 1, 0, 2, 1, 0, 6, 4, 1, 0, 24, 16, 6, 1, 0, 120, 72, 30, 8, 1, 0, 720, 372, 152, 48, 10, 1, 0, 5040, 2208, 828, 272, 70, 12, 1, 0, 40320, 14976, 4968, 1576, 440, 96, 14, 1, 0, 362880, 115200, 33192, 9696, 2720, 664, 126, 16, 1, 0, 3628800, 996480, 247968
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| T(n,k) is the number of lists of k unlabeled permutations whose total length is n. Unlabeled means each permutation is on an initial segment of the positive integers. Example: with dashes separating permutations, T(3,2) = 4 counts 1-12, 1-21, 12-1, 21-1. - David Callan (callan(AT)stat.wisc.edu), Nov 29 2007
For n > 0 Sum((-1)^i*row[n][i],i=0..n) is the number of indecomposable permutations A003319. [From Peter Luschny (peter(AT)luschny.de), Mar 13 2009]
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 171, #34.
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FORMULA
| T(n, k) = T(n-1, k-1)+ ((n+k-1)/k)*T(n-1, k); T(0, 0)=1, T(n, 0)=0 if n>0, T(0, k)=0 if k>0 . G.f. for the k-th column: (Sum_{i>=1} i!*t^i)^k = Sum_{n>=k} T(n, k)*t^n.
Sum_{k=0..n} T(n, k)*binomial(m, k) = A084938(m+n, m) . - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 31 2004
T(n, k) = Sum_{j>=0} A090753(j)*T(n-1, k+j-1). - DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Feb 18 2004
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EXAMPLE
| Triangle begins:
1;
0, 1;
0, 2, 1;
0, 6, 4, 1;
0, 24, 16, 6, 1;
0, 120, 72, 30, 8, 1;
0, 720, 372, 152, 48, 12, 1;
0, 5040, 2208, 828, 272, 70, 12, 1;
0, 40320, 14976, 4968, 1576, 440, 96, 14, 1;
0, 366880, 115200, 33192, 9696, 2720, 664, 126, 16, 1;
0, 3628800, 996480, 247968, 64704, 64704, 17312, 4380, 952, 160, 18, 1 ;...
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CROSSREFS
| Diagonals: A000007, A000142, A059371, A000012, A005843, A054000 . Row sums: A051296 . Another version: A059369.
Sequence in context: A205813 A127631 A122538 * A047922 A021830 A111184
Adjacent sequences: A090235 A090236 A090237 * A090239 A090240 A090241
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KEYWORD
| easy,nonn,tabl
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AUTHOR
| DELEHAM Philippe (kolotoko(AT)wanadoo.fr), Jan 23 2004, Jun 14 2007
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