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