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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. 2
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, 121, 24, 5, 1, 81, 1884, 2176, 578, 112, 25, 5, 1, 220, 5265, 7935, 2649, 580, 116, 25, 5, 1, 608, 14747, 28022, 11827, 2825, 602, 124, 25, 5, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
The consecutive patterns 101010, 101100, 110010, 110100, 111000 are counted. Here 1=Up=(1,1), 0=Down=(1,-1).
LINKS
EXAMPLE
T(3,1) = 5: 101010, 101100, 110010, 110100, 111000.
T(4,0) = 1: 11001100.
T(4,2) = 2: 10101010, 10110010.
T(5,0) = 1: 1110011000.
T(6,3) = 7: 101010101100, 101010110010, 101100101010, 101100101100, 110010101010, 110010110010, 110101010100.
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 0, 5;
: 4 : 1, 11, 2;
: 5 : 1, 33, 7, 1;
: 6 : 4, 90, 30, 7, 1;
: 7 : 11, 245, 142, 24, 6, 1;
: 8 : 29, 680, 570, 121, 24, 5, 1;
: 9 : 81, 1884, 2176, 578, 112, 25, 5, 1;
: 10 : 220, 5265, 7935, 2649, 580, 116, 25, 5, 1;
MAPLE
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, expand(add(b(x-1, y-1+2*j, irem(2*t+j, 32))*
`if`(2*t+j in {42, 44, 50, 52, 56}, z, 1), j=0..1))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 0)):
seq(T(n), n=0..14);
MATHEMATICA
b[x_, y_, t_] := b[x, y, t] = If[y < 0 || y > x, 0,
If[x == 0, 1, Expand[Sum[b[x-1, y-1 + 2j, Mod[2t+j, 32]]*
Switch[2t+j, 42|44|50|52|56, z, _, 1], {j, 0, 1}]]]];
T[n_] := CoefficientList[b[2n, 0, 0], z];
Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Apr 30 2022, after Alois P. Heinz *)
CROSSREFS
Column k=0 gives A243986.
Row sums give A000108.
Sequence in context: A370896 A307438 A198926 * A290395 A082974 A167635
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 17 2014
STATUS
approved

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Last modified June 21 18:40 EDT 2024. Contains 373556 sequences. (Running on oeis4.)