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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, k<=0<=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 Alois P. Heinz, Rows n = 0..300, flattened 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 Columns k=0-10 give: A243870, A243871, A243872, A243873, A243874, A243875, A243876, A243877, A243878, A243879, A243880. Row sums give A000108. T(738,k) = A243752(738,k). T(n,0) = A243753(n,738). Cf. A243882. Sequence in context: A007463 A159308 A163189 * A225691 A116846 A080558 Adjacent sequences:  A243878 A243879 A243880 * A243882 A243883 A243884 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Jun 13 2014 STATUS approved

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Last modified December 13 01:23 EST 2019. Contains 329963 sequences. (Running on oeis4.)