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A242450 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UUDDUDUUUUDUDDDDUUDD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-2)/8)), read by rows.

%I A242450 #44 Mar 27 2021 14:43:47
%S A242450 1,1,2,5,14,42,132,429,1430,4862,16795,1,58783,3,208002,10,742865,35,
%T A242450 2674314,126,9694383,462,35355954,1716,129638355,6435,477614391,24308,
%U A242450 1,1767170815,92372,3,6563767715,352694,11,24464914983,1351996,41,91477363496,5199988
%N A242450 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UUDDUDUUUUDUDDDDUUDD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-2)/8)), read by rows.
%C A242450 UUDDUDUUUUDUDDDDUUDD is a Dyck path that contains all sixteen consecutive step patterns of length 4.
%H A242450 Alois P. Heinz, <a href="/A242450/b242450.txt">Rows n = 0..500, flattened</a>
%e A242450 Triangle T(n,k) begins:
%e A242450 :  0 :           1;
%e A242450 :  1 :           1;
%e A242450 :  2 :           2;
%e A242450 :  3 :           5;
%e A242450 :  4 :          14;
%e A242450 :  5 :          42;
%e A242450 :  6 :         132;
%e A242450 :  7 :         429;
%e A242450 :  8 :        1430;
%e A242450 :  9 :        4862;
%e A242450 : 10 :       16795,       1;
%e A242450 : 11 :       58783,       3;
%e A242450 : 12 :      208002,      10;
%e A242450 : 13 :      742865,      35;
%e A242450 : 14 :     2674314,     126;
%e A242450 : 15 :     9694383,     462;
%e A242450 : 16 :    35355954,    1716;
%e A242450 : 17 :   129638355,    6435;
%e A242450 : 18 :   477614391,   24308,  1;
%e A242450 : 19 :  1767170815,   92372,  3;
%e A242450 : 20 :  6563767715,  352694, 11;
%e A242450 : 21 : 24464914983, 1351996, 41;
%p A242450 b:= proc(x, y, t) option remember; `if`(x=0, 1,
%p A242450      expand(`if`(y>=x-1, 0, b(x-1, y+1, [2, 3, 3, 2, 6, 3,
%p A242450        8, 9, 10, 11, 3, 13, 3, 2, 2, 2, 18, 19, 3, 2][t]))+
%p A242450      `if`(t=20, z, 1)*`if`(y=0, 0, b(x-1, y-1, [1, 1, 4, 5, 1, 7,
%p A242450        1, 1, 4, 4, 12, 5, 14, 15, 16, 17, 1, 1, 20, 5][t]))))
%p A242450     end:
%p A242450 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
%p A242450 seq(T(n), n=0..30);
%t A242450 b[x_, y_, t_] := b[x, y, t] = If[x == 0, 1, Expand[If[y >= x - 1, 0, b[x - 1, y + 1, {2, 3, 3, 2, 6, 3, 8, 9, 10, 11, 3, 13, 3, 2, 2, 2, 18, 19, 3, 2}[[t]]]] + If[t == 20, z, 1]*If[y == 0, 0, b[x - 1, y - 1, {1, 1, 4, 5, 1, 7, 1, 1, 4, 4, 12, 5, 14, 15, 16, 17, 1, 1, 20, 5}[[t]]]]]];
%t A242450 T[n_] := CoefficientList[b[2n, 0, 1], z];
%t A242450 T /@ Range[0, 30] // Flatten (* _Jean-François Alcover_, Mar 27 2021, after _Alois P. Heinz_ *)
%Y A242450 Row sums give A000108.
%Y A242450 T(834828,k) = A243752(834828,k).
%Y A242450 T(n,0) = A243753(n,834828).
%Y A242450 Cf. A243820.
%K A242450 nonn,tabf
%O A242450 0,3
%A A242450 _Alois P. Heinz_, Jun 12 2014