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A303160 Number of permutations p of [n] such that 0p has exactly ceiling(n/2) alternating runs.

%I A303160 #12 Aug 31 2021 07:42:02
%S A303160 1,1,1,3,7,43,148,1344,6171,74211,425976,6384708,43979902,789649750,
%T A303160 6346283560,132789007200,1219725741715,29145283614115,301190499710320,
%U A303160 8092186932120060,92921064554444490,2772830282722806978,35025128774218944648,1149343084932146388144
%N A303160 Number of permutations p of [n] such that 0p has exactly ceiling(n/2) alternating runs.
%H A303160 Alois P. Heinz, <a href="/A303160/b303160.txt">Table of n, a(n) for n = 0..456</a>
%F A303160 a(n) = A186370(n,ceiling(n/2)).
%e A303160 a(2) = 1: 12.
%e A303160 a(3) = 3: 132, 231, 321.
%e A303160 a(4) = 7: 1243, 1342, 1432, 2341, 2431, 3421, 4321.
%p A303160 b:= proc(n, k) option remember; `if`(k=0,
%p A303160       `if`(n=0, 1, 0), `if`(k<0 or k>n, 0,
%p A303160        k*b(n-1, k)+b(n-1, k-1)+(n-k+1)*b(n-1, k-2)))
%p A303160     end:
%p A303160 a:= n-> b(n, ceil(n/2)):
%p A303160 seq(a(n), n=0..25);
%t A303160 b[n_, k_] := b[n, k] = If[k == 0,
%t A303160      If[n == 0, 1, 0], If[k < 0 || k > n, 0,
%t A303160      k*b[n-1, k] + b[n-1, k-1] + (n-k+1)*b[n-1, k-2]]];
%t A303160 a[n_] := b[n, Ceiling[n/2]];
%t A303160 Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, Aug 31 2021, after _Alois P. Heinz_ *)
%Y A303160 Bisections give: A291677 (even part), A303159 (odd part).
%Y A303160 Cf. A186370.
%K A303160 nonn
%O A303160 0,4
%A A303160 _Alois P. Heinz_, Apr 19 2018