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A000349 One-half the number of permutations of length n with exactly 2 rising or falling successions.
(Formerly M3932 N1617)
9
0, 0, 0, 1, 5, 24, 128, 835, 6423, 56410, 554306, 6016077, 71426225, 920484892, 12793635300, 190730117959, 3035659077083, 51371100102990, 920989078354838, 17437084517068465, 347647092476801301, 7280060180210901232, 159755491837445900120, 3665942433747225901707 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

(1/2) times number of permutations of 12...n such that exactly 2 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.

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

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

FORMULA

Coefficient of t^2 in S[n](t) defined in A002464, divided by 2.

Recurrence: (n-3)*(n-2)*(n-4)^3*a(n) = (n-3)*(n^4-9*n^3+23*n^2-4*n-29)*(n-4)*a(n-1) - (n-1)*(n^4-12*n^3+57*n^2-125*n+104)*(n-4)*a(n-2) - (n-2)*(n-1)*(n^4-15*n^3+83*n^2-198*n+169)*a(n-3) + (n-3)^3*(n-2)^2*(n-1)*a(n-4). - Vaclav Kotesovec, Aug 10 2013

a(n) ~ sqrt(2*Pi)*n^(n+1/2)/exp(n+2). - 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-> ceil(coeff(S(n), t, 2)/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, 2]/2]; Table[a[n], {n, 0, 25}] (* Jean-Fran├žois Alcover, Mar 11 2014, after Alois P. Heinz *)

CROSSREFS

Cf. A002464, A000130, A086852. Equals A086853/2. A diagonal of A010028.

Sequence in context: A271544 A275267 A278679 * A036919 A020067 A066118

Adjacent sequences:  A000346 A000347 A000348 * A000350 A000351 A000352

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified March 28 15:38 EDT 2017. Contains 284243 sequences.