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 A243366 Number T(n,k) of Dyck paths of semilength n having exactly k (possibly overlapping) occurrences of the consecutive steps UDUUDU (with U=(1,1), D=(1,-1)); triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2)-1), read by rows. 13
 1, 1, 2, 5, 13, 1, 37, 5, 112, 19, 1, 352, 70, 7, 1136, 259, 34, 1, 3742, 962, 149, 9, 12529, 3585, 627, 54, 1, 42513, 13399, 2584, 279, 11, 145868, 50201, 10529, 1334, 79, 1, 505234, 188481, 42606, 6092, 474, 13, 1764157, 709001, 171563, 27048, 2561, 109, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Conjecture: Generally, column k is asymptotic to c(k) * d^n * n^(k-3/2), where d = 3.8821590268628506747194368909643384... is the root of the equation d^8 - 2*d^7 - 10*d^6 + 12*d^5 - 5*d^4 - 2*d^3 - 5*d^2 - 8*d - 3 = 0, and c(k) are specific constants (independent on n). - Vaclav Kotesovec, Jun 05 2014 LINKS Alois P. Heinz, Rows n = 0..200, flattened EXAMPLE T(4,1) = 1: UDUUDUDD. T(5,1) = 5: UDUDUUDUDD, UDUUDUDDUD, UDUUDUDUDD, UDUUDUUDDD, UUDUUDUDDD. T(6,1) = 19: UDUDUDUUDUDD, UDUDUUDUDDUD, UDUDUUDUDUDD, UDUDUUDUUDDD, UDUUDUDDUDUD, UDUUDUDDUUDD, UDUUDUDUDDUD, UDUUDUDUDUDD, UDUUDUDUUDDD, UDUUDUUDDDUD, UDUUDUUDDUDD, UDUUDUUUDDDD, UUDDUDUUDUDD, UUDUDUUDUDDD, UUDUUDUDDDUD, UUDUUDUDDUDD, UUDUUDUDUDDD, UUDUUDUUDDDD, UUUDUUDUDDDD. T(6,2) = 1: UDUUDUUDUDDD. T(7,2) = 7: UDUDUUDUUDUDDD, UDUUDUDUUDUDDD, UDUUDUUDUDDDUD, UDUUDUUDUDDUDD, UDUUDUUDUDUDDD, UDUUDUUDUUDDDD, UUDUUDUUDUDDDD. T(8,3) = 1: UDUUDUUDUUDUDDDD. Triangle T(n,k) begins: : 0 : 1; : 1 : 1; : 2 : 2; : 3 : 5; : 4 : 13, 1; : 5 : 37, 5; : 6 : 112, 19, 1; : 7 : 352, 70, 7; : 8 : 1136, 259, 34, 1; : 9 : 3742, 962, 149, 9; : 10 : 12529, 3585, 627, 54, 1; 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][t])* `if`(t=6, z, 1) +b(x-1, y-1, [1, 3, 1, 3, 6, 1][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, 2, 4}[[t]]]*If[t == 6, z, 1] + b[x-1, y-1, {1, 3, 1, 3, 6, 1}[[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, Feb 05 2015, after Alois P. Heinz *) CROSSREFS Column k=0-10 give: A243412, A243413, A243414, A243415, A243416, A243417, A243418, A243419, A243420, A243421, A243422. Row sums give A000108. T(n,floor(n/2)-1) gives A093178(n) for n>3. T(45,k) = A243752(45,k). T(n,0) = A243753(n,45). Sequence in context: A135331 A135329 A114508 * A139023 A241758 A173620 Adjacent sequences: A243363 A243364 A243365 * A243367 A243368 A243369 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Jun 03 2014 STATUS approved

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Last modified July 19 19:31 EDT 2024. Contains 374436 sequences. (Running on oeis4.)