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A243881 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUUDDDUD (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/4)), read by rows. 12
1, 1, 2, 5, 14, 41, 1, 129, 3, 419, 10, 1395, 35, 4737, 124, 1, 16338, 454, 4, 57086, 1684, 16, 201642, 6305, 65, 718855, 23781, 263, 1, 2583149, 90209, 1077, 5, 9346594, 343809, 4419, 23, 34023934, 1315499, 18132, 105, 124519805, 5050144, 74368, 472, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
UDUUUDDDUD is the only Dyck path of semilength 5 that contains all eight consecutive step patterns of length 3.
LINKS
EXAMPLE
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 5;
: 4 : 14;
: 5 : 41, 1;
: 6 : 129, 3;
: 7 : 419, 10;
: 8 : 1395, 35;
: 9 : 4737, 124, 1;
: 10 : 16338, 454, 4;
: 11 : 57086, 1684, 16;
: 12 : 201642, 6305, 65;
: 13 : 718855, 23781, 263, 1;
: 14 : 2583149, 90209, 1077, 5;
: 15 : 9346594, 343809, 4419, 23;
: 16 : 34023934, 1315499, 18132, 105;
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, 6, 2, 4, 2, 10, 2][t])+`if`(t=10,
z, 1)*b(x-1, y-1, [1, 3, 1, 3, 3, 7, 8, 9, 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..20);
MATHEMATICA
b[x_, y_, t_] := b[x, y, t] = If[y<0 || y>x, 0, If[x==0, 1, Expand[b[x-1, y+1, {2, 2, 4, 5, 6, 2, 4, 2, 10, 2}[[t]]] + If[t==10, z, 1]*b[x-1, y-1, {1, 3, 1, 3, 3, 7, 8, 9, 1, 3}[[t]]]]]]; T[n_] := Function[{p}, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[2*n, 0, 1]]; Table[T[n], {n, 0, 20}] // Flatten (* Jean-François Alcover, Mar 31 2015, after Alois P. Heinz *)
CROSSREFS
Row sums give A000108.
T(738,k) = A243752(738,k).
T(n,0) = A243753(n,738).
Cf. A243882.
Sequence in context: A007463 A159308 A163189 * A358416 A225691 A116846
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Jun 13 2014
STATUS
approved

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Last modified March 28 17:42 EDT 2024. Contains 371254 sequences. (Running on oeis4.)