A000130 One-half the number of permutations of length n with exactly 1 rising or falling successions. (Formerly M1528 N0598)

0, 0, 1, 2, 5, 20, 115, 790, 6217, 55160, 545135, 5938490, 70686805, 912660508, 12702694075, 189579135710, 3019908731105, 51139445487680, 917345570926087, 17376071107513090, 346563420097249645, 7259714390232227300, 159352909727731210835, 3657569576966074846118

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

Alois P. Heinz, Table of n, a(n) for n = 0..200

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

Adjacent sequences: A000127 A000128 A000129 * A000131 A000132 A000133

Keyword

nonn

Author

N. J. A. Sloane

Status

approved