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A243881
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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.
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12
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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
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OFFSET
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0,3
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COMMENTS
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UDUUUDDDUD is the only Dyck path of semilength 5 that contains all eight consecutive step patterns of length 3.
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LINKS
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EXAMPLE
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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;
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MAPLE
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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);
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MATHEMATICA
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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 *)
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CROSSREFS
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Columns k=0-10 give: A243870, A243871, A243872, A243873, A243874, A243875, A243876, A243877, A243878, A243879, A243880.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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