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A243998 Number T(n,k) of Dyck paths of semilength n with exactly k (possibly overlapping) occurrences of some of the consecutive patterns of Dyck paths of semilength 3; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

%I A243998 #17 Apr 30 2022 12:57:00
%S A243998 1,1,2,0,5,1,11,2,1,33,7,1,4,90,30,7,1,11,245,142,24,6,1,29,680,570,
%T A243998 121,24,5,1,81,1884,2176,578,112,25,5,1,220,5265,7935,2649,580,116,25,
%U A243998 5,1,608,14747,28022,11827,2825,602,124,25,5,1
%N A243998 Number T(n,k) of Dyck paths of semilength n with exactly k (possibly overlapping) occurrences of some of the consecutive patterns of Dyck paths of semilength 3; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.
%C A243998 The consecutive patterns 101010, 101100, 110010, 110100, 111000 are counted.  Here 1=Up=(1,1), 0=Down=(1,-1).
%H A243998 Alois P. Heinz, <a href="/A243998/b243998.txt">Rows n = 0..142, flattened</a>
%e A243998 T(3,1) = 5: 101010, 101100, 110010, 110100, 111000.
%e A243998 T(4,0) = 1: 11001100.
%e A243998 T(4,2) = 2: 10101010, 10110010.
%e A243998 T(5,0) = 1: 1110011000.
%e A243998 T(6,3) = 7: 101010101100, 101010110010, 101100101010, 101100101100, 110010101010, 110010110010, 110101010100.
%e A243998 Triangle T(n,k) begins:
%e A243998 :  0 :   1;
%e A243998 :  1 :   1;
%e A243998 :  2 :   2;
%e A243998 :  3 :   0,    5;
%e A243998 :  4 :   1,   11,    2;
%e A243998 :  5 :   1,   33,    7,    1;
%e A243998 :  6 :   4,   90,   30,    7,   1;
%e A243998 :  7 :  11,  245,  142,   24,   6,   1;
%e A243998 :  8 :  29,  680,  570,  121,  24,   5,  1;
%e A243998 :  9 :  81, 1884, 2176,  578, 112,  25,  5, 1;
%e A243998 : 10 : 220, 5265, 7935, 2649, 580, 116, 25, 5, 1;
%p A243998 b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
%p A243998       `if`(x=0, 1, expand(add(b(x-1, y-1+2*j, irem(2*t+j, 32))*
%p A243998       `if`(2*t+j in {42, 44, 50, 52, 56}, z, 1), j=0..1))))
%p A243998     end:
%p A243998 T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 0)):
%p A243998 seq(T(n), n=0..14);
%t A243998 b[x_, y_, t_] := b[x, y, t] = If[y < 0 || y > x, 0,
%t A243998      If[x == 0, 1, Expand[Sum[b[x-1, y-1 + 2j, Mod[2t+j, 32]]*
%t A243998      Switch[2t+j, 42|44|50|52|56, z, _, 1], {j, 0, 1}]]]];
%t A243998 T[n_] := CoefficientList[b[2n, 0, 0], z];
%t A243998 Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Apr 30 2022, after _Alois P. Heinz_ *)
%Y A243998 Column k=0 gives A243986.
%Y A243998 Row sums give A000108.
%Y A243998 Cf. A014486, A063171, A243966.
%K A243998 nonn,tabf
%O A243998 0,3
%A A243998 _Alois P. Heinz_, Jun 17 2014