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A303159 Number of permutations p of [2n+1] such that 0p has exactly n+1 alternating runs.

%I A303159 #14 Dec 21 2020 11:56:35
%S A303159 1,3,43,1344,74211,6384708,789649750,132789007200,29145283614115,
%T A303159 8092186932120060,2772830282722806978,1149343084932146388144,
%U A303159 566844242187778610648334,328043720353943611689811272,220147053200818211779539712908,169580070210721829547034445169024
%N A303159 Number of permutations p of [2n+1] such that 0p has exactly n+1 alternating runs.
%H A303159 Alois P. Heinz, <a href="/A303159/b303159.txt">Table of n, a(n) for n = 0..227</a>
%F A303159 a(n) = A186370(2n+1,n+1).
%F A303159 a(n) ~ c * d^n * n!^2, where d = 3.4210546206711870249402157940795853513031388... and c = 0.974460718185930534652526741942010711752... - _Vaclav Kotesovec_, Apr 29 2018
%e A303159 a(1) = 3: 132, 231, 321.
%p A303159 b:= proc(n, k) option remember; `if`(k=0,
%p A303159       `if`(n=0, 1, 0), `if`(k<0 or k>n, 0,
%p A303159        k*b(n-1, k)+b(n-1, k-1)+(n-k+1)*b(n-1, k-2)))
%p A303159     end:
%p A303159 a:= n-> b(2*n+1, n+1):
%p A303159 seq(a(n), n=0..20);
%t A303159 b[n_, k_] := b[n, k] = If[k == 0,
%t A303159     If[n == 0, 1, 0], If[k < 0 || k > n, 0,
%t A303159     k b[n-1, k] + b[n-1, k-1] + (n-k+1) b[n-1, k-2]]];
%t A303159 a[n_] := b[2n+1, n+1];
%t A303159 a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 21 2020, after _Alois P. Heinz_ *)
%Y A303159 Bisection (odd part) of A303160.
%Y A303159 Cf. A186370.
%K A303159 nonn
%O A303159 0,2
%A A303159 _Alois P. Heinz_, Apr 19 2018