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 A230797 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step pattern up, down, up, down; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-3)/2)), read by rows. 6
 1, 1, 2, 6, 24, 104, 16, 528, 192, 3296, 1472, 272, 23168, 12800, 4352, 179712, 132352, 42880, 7936, 1573632, 1366016, 530432, 158720, 15207424, 14781952, 7662336, 1911296, 353792, 158880768, 178102272, 101713920, 31813632, 8491008, 1801996288, 2282645504 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Rows n = 0..170, flattened EXAMPLE T(5,1) = 16: 13254, 14253, 14352, 15243, 15342, 23154, 24153, 24351, 25143, 25341, 34152, 34251, 35142, 35241, 45132, 45231. T(7,2) = 272: 1325476, 1326475, 1326574, ..., 6735241, 6745132, 6745231. Triangle T(n,k) begins: : 0 : 1; : 1 : 1; : 2 : 2; : 3 : 6; : 4 : 24; : 5 : 104, 16; : 6 : 528, 192; : 7 : 3296, 1472, 272; : 8 : 23168, 12800, 4352; : 9 : 179712, 132352, 42880, 7936; : 10 : 1573632, 1366016, 530432, 158720; MAPLE b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand( add(b(u-j, o+j-1, [1, 3, 1, 3][t])*`if`(t=4, x, 1), j=1..u)+ add(b(u+j-1, o-j, [2, 2, 4, 2][t]), j=1..o))) end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)): seq(T(n), n=0..15); # Alois P. Heinz, Oct 30 2013 MATHEMATICA b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[Sum[b[u-j, o+j-1, {1, 3, 1, 3}[[t]]]*If[t == 4, x, 1], {j, 1, u}] + Sum[b[u+j-1, o-j, {2, 2, 4, 2}[[t]]], {j, 1, o}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-François Alcover, Oct 24 2016, after Alois P. Heinz *) CROSSREFS Columns k=0-2 give: A177520, A230832, A264077. T(2n-1,n-2) gives A000182(n) for n>=3. Row sums give: A000142. Cf. A242783, A242784, A295987. Sequence in context: A078486 A352364 A129817 * A128652 A152316 A177520 Adjacent sequences: A230794 A230795 A230796 * A230798 A230799 A230800 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Oct 30 2013 STATUS approved

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Last modified August 10 16:53 EDT 2024. Contains 375058 sequences. (Running on oeis4.)