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A243966 Number of Dyck paths of semilength n such that all five consecutive patterns of Dyck paths of semilength 3 occur at least once.

%I A243966 #16 Jan 26 2021 16:20:39
%S A243966 0,0,0,0,0,0,0,0,0,0,0,0,12,138,1152,8166,52098,308964,1733444,
%T A243966 9311300,48280464,243112106,1194286106,5744306228,27129749648,
%U A243966 126111332862,578106334718,2617667137358,11723920607410,51998857149406,228621028644376,997286152915772
%N A243966 Number of Dyck paths of semilength n such that all five consecutive patterns of Dyck paths of semilength 3 occur at least once.
%C A243966 The five consecutive patterns that occur at least once each are 101010, 101100, 110010, 110100, 111000.  Here 1=Up=(1,1), 0=Down=(1,-1).
%H A243966 Alois P. Heinz, <a href="/A243966/b243966.txt">Table of n, a(n) for n = 0..300</a>
%e A243966 a(12) = 12: 101010110010110100111000, 101010110010111000110100, 101100101010110100111000, 101100101010111000110100, 110100101010110010111000, 110100101100101010111000, 110100111000101010110010, 110100111000101100101010, 111000101010110010110100, 111000101100101010110100, 111000110100101010110010, 111000110100101100101010.
%e A243966 Here 1=Up=(1,1), 0=Down=(1,-1).
%p A243966 b:= proc(x, y, l) option remember; local m; m:= min(l[]);
%p A243966       `if`(y>x or y<0 or 7-m>x, 0, `if`(x=0, 1,
%p A243966       b(x-1, y+1, [[2, 3, 4, 4, 2, 2, 7][l[1]],
%p A243966        [2, 3, 3, 5, 3, 2, 7][l[2]], [2, 3, 3, 2, 6, 3,7][l[3]],
%p A243966        [2, 2, 4, 5, 2, 4, 7][l[4]], [2, 2, 4, 2, 6, 2,7][l[5]]])+
%p A243966       b(x-1, y-1, [[1, 1, 1, 5, 6, 7, 7][l[1]],
%p A243966        [1, 1, 4, 1, 6, 7, 7][l[2]], [1, 1, 4, 5, 1, 7, 7][l[3]],
%p A243966        [1, 3, 1, 3, 6, 7, 7][l[4]], [1, 3, 1, 5, 1, 7, 7][l[5]]])))
%p A243966     end:
%p A243966 a:= n-> b(2*n, 0, [1$5]):
%p A243966 seq(a(n), n=0..35);
%t A243966 b[x_, y_, l_] := b[x, y, l] = Module[{m = Min[l]},
%t A243966         If[y>x || y<0 || 7-m>x, 0, If[x == 0, 1,
%t A243966         b[x-1, y+1, MapIndexed[#1[[l[[#2[[1]] ]] ]]&,
%t A243966         {{2, 3, 4, 4, 2, 2, 7},
%t A243966          {2, 3, 3, 5, 3, 2, 7},
%t A243966          {2, 3, 3, 2, 6, 3, 7},
%t A243966          {2, 2, 4, 5, 2, 4, 7},
%t A243966          {2, 2, 4, 2, 6, 2, 7}}]]] +
%t A243966         b[x-1, y-1, MapIndexed[#1[[l[[#2[[1]] ]] ]]&,
%t A243966         {{1, 1, 1, 5, 6, 7, 7},
%t A243966          {1, 1, 4, 1, 6, 7, 7},
%t A243966          {1, 1, 4, 5, 1, 7, 7},
%t A243966          {1, 3, 1, 3, 6, 7, 7},
%t A243966          {1, 3, 1, 5, 1, 7, 7}}]]]];
%t A243966 a[n_] := b[2n, 0, {1, 1, 1, 1, 1}];
%t A243966 a /@ Range[0, 35] (* _Jean-François Alcover_, Jan 26 2021, after _Alois P. Heinz_ *)
%Y A243966 Cf. A014486, A063171, A243820, A243965, A243986.
%K A243966 nonn
%O A243966 0,13
%A A243966 _Alois P. Heinz_, Jun 16 2014