A001266 One-half the number of permutations of length n without rising or falling successions. (Formerly M4426 N1871)

0, 0, 1, 7, 45, 323, 2621, 23811, 239653, 2648395, 31889517, 415641779, 5830753109, 87601592187, 1403439027805, 23883728565283, 430284458893701, 8181419271349931, 163730286973255373, 3440164703027845395, 75718273707281368117, 1742211593431076483419

Offset

2,4

Comments

(1/2) times number of permutations of 12...n such that none of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

a(n) is also the number of Hamiltonian paths in the n-path complement graph. - Eric W. Weisstein, Apr 11 2018

References

F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 263.

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

Seiichi Manyama, Table of n, a(n) for n = 2..450 (first 199 terms from Alois P. Heinz)

J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.

Eric Weisstein's World of Mathematics, Hamiltonian Path

Eric Weisstein's World of Mathematics, Path Complement Graph

Formula

a(n) = A002464(n)/2.

(1/2) times coefficient of t^0 in S[n](t) defined in A002464.

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, 0)/2:
seq(a(n), n=2..25);  # Alois P. Heinz, Jan 11 2013

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, 0]/2; Table[a[n], {n, 2, 25}] (* Jean-François Alcover, Mar 24 2014, after Alois P. Heinz *)
CoefficientList[Series[((Exp[(1 + x)/((-1 + x) x)] (1 + x) Gamma[0, (1 + x)/((-1 + x) x)])/((-1 + x) x) - x - 1)/(2 x), {x, 0, 20}], x] (* Eric W. Weisstein, Apr 11 2018 *)
RecurrenceTable[{a[n] == (n + 1) a[n - 1] - (n - 2) a[n - 2] - (n - 5) a[n - 3] + (n - 3) a[n - 4], a[0] == a[1] == 1/2,
a[2] == a[3] == 0}, a, {n, 2, 20}] (* Eric W. Weisstein, Apr 11 2018 *)

Crossrefs

Sequence A002464 divided by 2 for n >= 2. A diagonal of A010028.

Sequence in context: A103719 A134437 A018927 * A071971 A370253 A337553

Adjacent sequences: A001263 A001264 A001265 * A001267 A001268 A001269

Keyword

nonn

Author

N. J. A. Sloane

Extensions

More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 16 2001

Status

approved