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 A001266 One-half the number of permutations of length n without rising or falling successions. (Formerly M4426 N1871) 6
 0, 0, 1, 7, 45, 323, 2621, 23811, 239653, 2648395, 31889517, 415641779, 5830753109, 87601592187, 1403439027805, 23883728565283, 430284458893701, 8181419271349931, 163730286973255373, 3440164703027845395, 75718273707281368117, 1742211593431076483419 (list; graph; refs; listen; history; text; internal format)
 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 Alois P. Heinz and 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 A337553 A006680 Adjacent sequences:  A001263 A001264 A001265 * A001267 A001268 A001269 KEYWORD nonn AUTHOR EXTENSIONS More terms from Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Feb 16 2001 STATUS approved

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Last modified January 23 03:02 EST 2021. Contains 340384 sequences. (Running on oeis4.)