A086854 - OEIS

%I #13 Apr 09 2014 09:51:19

%S 0,0,0,0,2,16,120,888,7198,64968,650644,7165200,86059242,1119549472,

%T 15682257872,235336043976,3766695159030,64052134910168,

%U 1153211148654348,21915344800505888,438380075974889154,9207290871553008240,202585136417883766472,4659950328485470292632

%N Number of permutations of length n with exactly 3 rising or falling successions.

%C Permutations of 12...n such that exactly 3 of the following occur: 12, 23, ..., (n-1)n, 21, 32, ..., n(n-1).

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

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

%H Alois P. Heinz, <a href="/A086854/b086854.txt">Table of n, a(n) for n = 0..200</a>

%F Coefficient of t^3 in S[n](t) defined in A002464.

%F Recurrence (for n>4): (n-4)*(2*n^6 - 52*n^5 + 557*n^4 - 3136*n^3 + 9740*n^2 - 15727*n + 10242)*a(n) = (n-4)*(2*n^7 - 50*n^6 + 511*n^5 - 2693*n^4 + 7450*n^3 - 9041*n^2 - 157*n + 6666)*a(n-1) - (2*n^8 - 58*n^7 + 735*n^6 - 5289*n^5 + 23430*n^4 - 64575*n^3 + 106105*n^2 - 92312*n + 30900)*a(n-2) - (2*n^7 - 54*n^6 + 615*n^5 - 3795*n^4 + 13554*n^3 - 27681*n^2 + 29473*n - 12330)*(n-2)*a(n-3) + (2*n^6 - 40*n^5 + 327*n^4 - 1388*n^3 + 3184*n^2 - 3675*n + 1626)*(n-2)^2*a(n-4). - _Vaclav Kotesovec_, Aug 11 2013

%F a(n) ~ 4/3*exp(-2) * n! = n! * 0.45231366335478... - _Vaclav Kotesovec_, Aug 11 2013

%p S:= proc(n) option remember; `if`(n<4, [1, 1, 2*t, 4*t+2*t^2]

%p        [n+1], expand((n+1-t)*S(n-1) -(1-t)*(n-2+3*t)*S(n-2)

%p        -(1-t)^2*(n-5+t)*S(n-3) +(1-t)^3*(n-3)*S(n-4)))

%p     end:

%p a:= n-> coeff(S(n), t, 3):

%p seq(a(n), n=0..25);  # _Alois P. Heinz_, Jan 11 2013

%t 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, 3]; Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Apr 09 2014, after _Alois P. Heinz_ *)

%Y Cf. A002464, A000130, A000349, A001267, A086852, A086853. A diagonal of A001100.

%K nonn

%O 0,5

%A _N. J. A. Sloane_, Aug 19 2003