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A000130 One-half the number of permutations of length n with exactly 1 rising or falling successions.
(Formerly M1528 N0598)
12
0, 0, 1, 2, 5, 20, 115, 790, 6217, 55160, 545135, 5938490, 70686805, 912660508, 12702694075, 189579135710, 3019908731105, 51139445487680, 917345570926087, 17376071107513090, 346563420097249645, 7259714390232227300, 159352909727731210835, 3657569576966074846118 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
(1/2) times number of permutations of 12...n such that exactly one of the following occurs: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
Partial sums seem to be in A000239. - Ralf Stephan, Aug 28 2003
REFERENCES
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).
LINKS
FORMULA
Coefficient of t^1 in S[n](t) defined in A002464, divided by 2.
a(n) ~ exp(-2) * n!. - Vaclav Kotesovec, Sep 11 2014
MAPLE
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, 1)/2:
seq(a(n), n=0..30); # Alois P. Heinz, Dec 21 2012
MATHEMATICA
S[n_] := S[n] = 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]]]; a[n_] := Coefficient[S[n], t, 1]/2; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
CROSSREFS
Cf. A002464, A086853. Equals A086852/2. A diagonal of A010028.
Sequence in context: A127065 A168357 A052850 * A288841 A009599 A112833
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified May 29 11:52 EDT 2024. Contains 372940 sequences. (Running on oeis4.)