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A001267
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One-half the number of permutations of length n with exactly 3 rising or falling successions.
(Formerly M4550 N1934)
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8
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0, 0, 0, 0, 1, 8, 60, 444, 3599, 32484, 325322, 3582600, 43029621, 559774736, 7841128936, 117668021988, 1883347579515, 32026067455084, 576605574327174, 10957672400252944, 219190037987444577, 4603645435776504120, 101292568208941883236, 2329975164242735146316
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OFFSET
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0,6
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COMMENTS
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(1/2) times number of permutations of 12...n such that exactly 3 of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
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REFERENCES
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F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.
J. Riordan, A recurrence for permutations without rising or falling successions. Ann. Math. Statist. 36 (1965), 708-710.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..200
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FORMULA
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Coefficient of t^3 in S[n](t) defined in A002464, divided by 2.
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MAPLE
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S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]
[n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)
-(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))
end:
a:= n-> coeff(S(n), t, 3)/2:
seq(a(n), n=0..25); # Alois P. Heinz, Jan 11 2013
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CROSSREFS
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Cf. A002464, A000130, A086852. Equals A086854/2. A diagonal of A010028.
Sequence in context: A093132 A094169 A129325 * A099156 A199526 A129331
Adjacent sequences: A001264 A001265 A001266 * A001268 A001269 A001270
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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