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 A001267 One-half the number of permutations of length n with exactly 3 rising or falling successions. (Formerly M4550 N1934) 8
 0, 0, 0, 0, 1, 8, 60, 444, 3599, 32484, 325322, 3582600, 43029621, 559774736, 7841128936, 117668021988, 1883347579515, 32026067455084, 576605574327174, 10957672400252944, 219190037987444577, 4603645435776504120, 101292568208941883236, 2329975164242735146316 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS (1/2) times number of permutations of 12...n such that exactly 3 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^3 in S[n](t) defined in A002464, divided by 2. a(n) ~ 2/(3*exp(2)) * n!. - Vaclav Kotesovec, Aug 10 2013 Recurrence: (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 10 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-> coeff(S(n), t, 3)/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_] := Coefficient[S[n], t, 3]/2; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *) CROSSREFS Cf. A002464, A000130, A086852. Equals A086854/2. A diagonal of A010028. Sequence in context: A094169 A129325 A285391 * A099156 A245391 A254658 Adjacent sequences:  A001264 A001265 A001266 * A001268 A001269 A001270 KEYWORD nonn AUTHOR STATUS approved

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Last modified April 8 01:57 EDT 2020. Contains 333312 sequences. (Running on oeis4.)