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%I #18 Mar 27 2021 15:15:24
%S 1,1,2,5,14,41,1,129,3,419,10,1395,35,4737,124,1,16338,454,4,57086,
%T 1684,16,201642,6305,65,718855,23781,263,1,2583149,90209,1077,5,
%U 9346594,343809,4419,23,34023934,1315499,18132,105,124519805,5050144,74368,472,1
%N Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/4)), read by rows.
%C UDUUUDDDUD is the only Dyck path of semilength 5 that contains all eight consecutive step patterns of length 3.
%H Alois P. Heinz, <a href="/A243881/b243881.txt">Rows n = 0..300, flattened</a>
%e Triangle T(n,k) begins:
%e : 0 : 1;
%e : 1 : 1;
%e : 2 : 2;
%e : 3 : 5;
%e : 4 : 14;
%e : 5 : 41, 1;
%e : 6 : 129, 3;
%e : 7 : 419, 10;
%e : 8 : 1395, 35;
%e : 9 : 4737, 124, 1;
%e : 10 : 16338, 454, 4;
%e : 11 : 57086, 1684, 16;
%e : 12 : 201642, 6305, 65;
%e : 13 : 718855, 23781, 263, 1;
%e : 14 : 2583149, 90209, 1077, 5;
%e : 15 : 9346594, 343809, 4419, 23;
%e : 16 : 34023934, 1315499, 18132, 105;
%p b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0, `if`(x=0, 1,
%p expand(b(x-1, y+1, [2, 2, 4, 5, 6, 2, 4, 2, 10, 2][t])+`if`(t=10,
%p z, 1)*b(x-1, y-1, [1, 3, 1, 3, 3, 7, 8, 9, 1, 3][t]))))
%p end:
%p T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
%p seq(T(n), n=0..20);
%t b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, 1, Expand[b[x-1, y+1, {2, 2, 4, 5, 6, 2, 4, 2, 10, 2}[[t]]] + If[t==10, z, 1]*b[x-1, y-1, {1, 3, 1, 3, 3, 7, 8, 9, 1, 3}[[t]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* _Jean-François Alcover_, Mar 31 2015, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A243870, A243871, A243872, A243873, A243874, A243875, A243876, A243877, A243878, A243879, A243880.
%Y Row sums give A000108.
%Y T(738,k) = A243752(738,k).
%Y T(n,0) = A243753(n,738).
%Y Cf. A243882.
%K nonn,tabf
%O 0,3
%A _Alois P. Heinz_, Jun 13 2014
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