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A008970
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Triangle T(n,k) = P(n,k)/2, n >= 2, 1<=k<n, of one-half of number of permutations of 1..n such that the differences have k runs with the same signs.
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11
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1, 1, 2, 1, 6, 5, 1, 14, 29, 16, 1, 30, 118, 150, 61, 1, 62, 418, 926, 841, 272, 1, 126, 1383, 4788, 7311, 5166, 1385, 1, 254, 4407, 22548, 51663, 59982, 34649, 7936, 1, 510, 13736, 100530, 325446, 553410, 517496, 252750, 50521, 1, 1022, 42236
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 2,3
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 261, #13, P_{n,k}.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 260, Table 7.2.1.
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LINKS
| M. Bona and R. Ehrenborg, [math/9902020] A combinatorial proof of the log-concavity of the numbers of permutations with k runs
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FORMULA
| Let P(n, k) = number of permutations of [1..n] with k "sequences". Note that A008970 gives P(n, k)/2. Then g.f.: Sum_{n, k} P(n, k)*u^k*t^n/n! = (1+u)^(-1)*((1-u)*(1-sin(v+t*cos(v))-1) where u = sin v.
P(n, 1)=2, P(n, k) = k*P(n-1, k) + 2*P(n-1, k-1) + (n-k)*P(n-1, k-2).
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EXAMPLE
| 1; 1,2; 1,6,5; 1,14,29,16; ...
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CROSSREFS
| Diagonals give A000352, A000486, A000506, A000111, A000708, A091303. A059427 gives triangle of P(n, k).
Sequence in context: A108767 A046817 A193817 * A055896 A193723 A159965
Adjacent sequences: A008967 A008968 A008969 * A008971 A008972 A008973
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KEYWORD
| tabl,nonn,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Feb 01 2001
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