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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.
2
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16795, 1, 58783, 3, 208002, 10, 742865, 35, 2674314, 126, 9694383, 462, 35355954, 1716, 129638355, 6435, 477614390, 24310, 1767170813, 92376, 1, 6563767708, 352708, 4, 24464914958, 1352046, 16, 91477363405, 5200170, 65
OFFSET
0,3
COMMENTS
UDUUDDUUUUDUDDDDUDUD is a Dyck path that contains all sixteen consecutive step patterns of length 4.
LINKS
EXAMPLE
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 5;
: 4 : 14;
: 5 : 42;
: 6 : 132;
: 7 : 429;
: 8 : 1430;
: 9 : 4862;
: 10 : 16795, 1;
: 11 : 58783, 3;
: 12 : 208002, 10;
: 13 : 742865, 35;
: 14 : 2674314, 126;
: 15 : 9694383, 462;
: 16 : 35355954, 1716;
: 17 : 129638355, 6435;
: 18 : 477614390, 24310;
: 19 : 1767170813, 92376, 1;
: 20 : 6563767708, 352708, 4;
: 21 : 24464914958, 1352046, 16;
MAPLE
b:= proc(x, y, t) option remember; `if`(y<0 or y>x, 0,
`if`(x=0, 1, expand(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) *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]))))
end:
T:= n-> (p-> seq(coeff(p, z, i), i=0..degree(p)))(b(2*n, 0, 1)):
seq(T(n), n=0..30);
MATHEMATICA
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[n_] := CoefficientList[b[2n, 0, 1], z];
T /@ Range[0, 30] // Flatten (* Jean-François Alcover, Mar 27 2021, after Alois P. Heinz *)
CROSSREFS
Row sums give A000108.
T(736522,k) = A243752(736522,k).
T(n,0) = A243753(n,736522).
Cf. A243820.
Sequence in context: A261591 A291823 A287972 * A242450 A211216 A261592
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 11 2014
STATUS
approved