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A246247 Number of permutations of [n] with exactly two occurrences of the consecutive step pattern up, down, down.

%I A246247 #15 Dec 28 2020 09:53:30
%S A246247 99,2214,38394,591543,8826246,131367258,1989555210,30951663300,
%T A246247 497599843140,8291940960690,143459287215300,2578465192541220,
%U A246247 48147387009459165,933704978071539690,18794023286090727870,392361396798154377681,8489006744706293477274
%N A246247 Number of permutations of [n] with exactly two occurrences of the consecutive step pattern up, down, down.
%H A246247 Alois P. Heinz and Vaclav Kotesovec, <a href="/A246247/b246247.txt">Table of n, a(n) for n = 7..300</a> (first 160 terms from Alois P. Heinz)
%F A246247 a(n) ~ c * (3*sqrt(3)/(2*Pi))^n * n! * n^2, where c = 0.10205535828170995196503... = c0 * (c0-1)^2 / 18, and c0 = (1 + exp(Pi/sqrt(3))) * sqrt(3) / (2*Pi). - _Vaclav Kotesovec_, Aug 22 2014
%p A246247 b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
%p A246247       add(b(u-j, o+j-1, [1, 3, 1][t])*`if`(t=3, x, 1), j=1..u)+
%p A246247       add(b(u+j-1, o-j, 2), j=1..o)))
%p A246247     end:
%p A246247 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
%p A246247 seq([T(n)][3], n=7..20); # _Vaclav Kotesovec_, Aug 22 2014 after Alois P. Heinz
%t A246247 b[u_, o_, t_] := b[u, o, t] = If[u + o == 0, 1, Expand[Sum[b[u - j, o + j - 1, {1, 3, 1}[[t]]]*If[t == 3, x, 1], {j, 1, u}] + Sum[b[u + j - 1, o - j, 2], {j, 1, o}]]];
%t A246247 a[n_] := Coefficient[b[n, 0, 1], x, 2];
%t A246247 a /@ Range[7, 20] (* _Jean-François Alcover_, Dec 28 2020, after Maple *)
%Y A246247 Column k=2 of A242819.
%K A246247 nonn
%O A246247 7,1
%A A246247 _Alois P. Heinz_, Aug 20 2014