OFFSET
0,7
COMMENTS
(1/2) times number of permutations of 12...n such that exactly 4 of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).
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
Alois P. Heinz, Table of n, a(n) for n = 0..200
J. Riordan, A recurrence for permutations without rising or falling successions, Ann. Math. Statist. 36 (1965), 708-710.
FORMULA
Coefficient of t^4 in S[n](t) defined in A002464, divided by 2.
Recurrence (for n>5): (n-5)*(n^8 - 41*n^7 + 730*n^6 - 7358*n^5 + 45799*n^4 - 179702*n^3 + 432498*n^2 - 581244*n + 332100)*a(n) = (n^10 - 45*n^9 + 895*n^8 - 10301*n^7 + 75340*n^6 - 361190*n^5 + 1124682*n^4 - 2150033*n^3 + 2147364*n^2 - 499899*n - 544266)*a(n-1) - (n^10 - 44*n^9 + 869*n^8 - 10112*n^7 + 76390*n^6 - 388742*n^5 + 1336932*n^4 - 3028095*n^3 + 4237931*n^2 - 3198426*n + 917988)*a(n-2) - (n^10 - 43*n^9 + 823*n^8 - 9195*n^7 + 66108*n^6 - 318138*n^5 + 1033118*n^4 - 2224673*n^3 + 3023402*n^2 - 2325285*n + 761190)*a(n-3) + (n^8 - 33*n^7 + 471*n^6 - 3783*n^5 + 18594*n^4 - 56865*n^3 + 104723*n^2 - 104847*n + 42783)*(n-2)^2*a(n-4). - Vaclav Kotesovec, Aug 11 2013
a(n) ~ n!*exp(-2)/3. - Vaclav Kotesovec, Aug 11 2013
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-> ceil(coeff(S(n), t, 4)/2):
seq(a(n), n=0..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_] := Ceiling[Coefficient[S[n], t, 4]/2]; Table [a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 24 2014, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved