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A303160 Number of permutations p of [n] such that 0p has exactly ceiling(n/2) alternating runs. 4
1, 1, 1, 3, 7, 43, 148, 1344, 6171, 74211, 425976, 6384708, 43979902, 789649750, 6346283560, 132789007200, 1219725741715, 29145283614115, 301190499710320, 8092186932120060, 92921064554444490, 2772830282722806978, 35025128774218944648, 1149343084932146388144 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..456

FORMULA

a(n) = A186370(n,ceiling(n/2)).

EXAMPLE

a(2) = 1: 12.

a(3) = 3: 132, 231, 321.

a(4) = 7: 1243, 1342, 1432, 2341, 2431, 3421, 4321.

MAPLE

b:= proc(n, k) option remember; `if`(k=0,

      `if`(n=0, 1, 0), `if`(k<0 or k>n, 0,

       k*b(n-1, k)+b(n-1, k-1)+(n-k+1)*b(n-1, k-2)))

    end:

a:= n-> b(n, ceil(n/2)):

seq(a(n), n=0..25);

MATHEMATICA

b[n_, k_] := b[n, k] = If[k == 0,

     If[n == 0, 1, 0], If[k < 0 || k > n, 0,

     k*b[n-1, k] + b[n-1, k-1] + (n-k+1)*b[n-1, k-2]]];

a[n_] := b[n, Ceiling[n/2]];

Table[a[n], {n, 0, 25}] (* Jean-Fran├žois Alcover, Aug 31 2021, after Alois P. Heinz *)

CROSSREFS

Bisections give: A291677 (even part), A303159 (odd part).

Cf. A186370.

Sequence in context: A101208 A050633 A107636 * A258435 A074268 A019026

Adjacent sequences:  A303157 A303158 A303159 * A303161 A303162 A303163

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Apr 19 2018

STATUS

approved

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Last modified November 27 20:49 EST 2021. Contains 349395 sequences. (Running on oeis4.)