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A243838 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUDDUUUUDUDDDDUDUD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/9)), read by rows.

%I A243838 #24 Mar 27 2021 14:43:00
%S A243838 1,1,2,5,14,42,132,429,1430,4862,16795,1,58783,3,208002,10,742865,35,
%T A243838 2674314,126,9694383,462,35355954,1716,129638355,6435,477614390,24310,
%U A243838 1767170813,92376,1,6563767708,352708,4,24464914958,1352046,16,91477363405,5200170,65
%N A243838 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUDDUUUUDUDDDDUDUD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/9)), read by rows.
%C A243838 UDUUDDUUUUDUDDDDUDUD is a Dyck path that contains all sixteen consecutive step patterns of length 4.
%H A243838 Alois P. Heinz, <a href="/A243838/b243838.txt">Rows n = 0..350, flattened</a>
%e A243838 Triangle T(n,k) begins:
%e A243838 :  0 :           1;
%e A243838 :  1 :           1;
%e A243838 :  2 :           2;
%e A243838 :  3 :           5;
%e A243838 :  4 :          14;
%e A243838 :  5 :          42;
%e A243838 :  6 :         132;
%e A243838 :  7 :         429;
%e A243838 :  8 :        1430;
%e A243838 :  9 :        4862;
%e A243838 : 10 :       16795,       1;
%e A243838 : 11 :       58783,       3;
%e A243838 : 12 :      208002,      10;
%e A243838 : 13 :      742865,      35;
%e A243838 : 14 :     2674314,     126;
%e A243838 : 15 :     9694383,     462;
%e A243838 : 16 :    35355954,    1716;
%e A243838 : 17 :   129638355,    6435;
%e A243838 : 18 :   477614390,   24310;
%e A243838 : 19 :  1767170813,   92376,  1;
%e A243838 : 20 :  6563767708,  352708,  4;
%e A243838 : 21 : 24464914958, 1352046, 16;
%p A243838 b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
%p A243838      `if`(x=0, 1, expand(b(x-1, y+1, [2, 2, 4, 5, 2, 4,
%p A243838        8, 9, 10, 11, 2, 13, 5, 4, 2, 2, 18, 2, 20, 5][t])
%p A243838       +`if`(t=20, z, 1) *b(x-1, y-1, [1, 3, 1, 3, 6, 7,
%p A243838        1, 3, 3, 3, 12, 1, 14, 15, 16, 17, 1, 19, 1, 3][t]))))
%p A243838     end:
%p A243838 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
%p A243838 seq(T(n), n=0..30);
%t A243838 b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Expand[If[y >= x - 1, 0, b[x - 1, y + 1, {2, 2, 4, 5, 2, 4, 8, 9, 10, 11, 2, 13, 5, 4, 2, 2, 18, 2, 20, 5}[[t]]]] + If[t == 20, z, 1]*If[y == 0, 0, b[x - 1, y - 1, {1, 3, 1, 3, 6, 7, 1, 3, 3, 3, 12, 1, 14, 15, 16, 17, 1, 19, 1, 3}[[t]]]]]];
%t A243838 T[n_] := CoefficientList[b[2n, 0, 1], z];
%t A243838 T /@ Range[0, 30] // Flatten (* _Jean-François Alcover_, Mar 27 2021, after _Alois P. Heinz_ *)
%Y A243838 Row sums give A000108.
%Y A243838 T(736522,k) = A243752(736522,k).
%Y A243838 T(n,0) = A243753(n,736522).
%Y A243838 Cf. A243820.
%K A243838 nonn,tabf
%O A243838 0,3
%A A243838 _Alois P. Heinz_, Jun 11 2014