A000130 One-half the number of permutations of length n with exactly 1 rising or falling successions.

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

Author

N. J. A. Sloane

Keywords

• nonn

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

Crossrefs

Cf. A002464, A086853. Equals A086852/2. A diagonal of A010028.

Programs

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 *)

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