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 A295987 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive step patterns 010 or 101, where 1=up and 0=down; triangle T(n,k), n >= 0, k = max(0, n-3), read by rows. 11
 1, 1, 2, 6, 14, 10, 52, 36, 32, 204, 254, 140, 122, 1010, 1368, 1498, 620, 544, 5466, 9704, 9858, 9358, 3164, 2770, 34090, 67908, 90988, 72120, 63786, 18116, 15872, 233026, 545962, 762816, 839678, 560658, 470262, 115356, 101042, 1765836, 4604360, 7458522 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Alois P. Heinz, Rows n = 0..143, flattened EXAMPLE Triangle T(n,k) begins: :      1; :      1; :      2; :      6; :     14,     10; :     52,     36,     32; :    204,    254,    140,    122; :   1010,   1368,   1498,    620,    544; :   5466,   9704,   9858,   9358,   3164,   2770; :  34090,  67908,  90988,  72120,  63786,  18116,  15872; : 233026, 545962, 762816, 839678, 560658, 470262, 115356, 101042; MAPLE b:= proc(u, o, t, h) option remember; expand(            `if`(u+o=0, 1, `if`(t=0, add(b(u-j, j-1, 1\$2), j=1..u),        add(`if`(h=3, x, 1)*b(u-j, o+j-1, [1, 3, 1][t], 2), j=1..u)+        add(`if`(t=3, x, 1)*b(u+j-1, o-j, 2, [1, 3, 1][h]), j=1..o))))     end: T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0\$3)): seq(T(n), n=0..12); MATHEMATICA b[u_, o_, t_, h_] := b[u, o, t, h] = Expand[If[u + o == 0, 1, If[t == 0, Sum[b[u - j, j - 1, 1, 1], {j, 1, u}], Sum[If[h == 3, x, 1]*b[u - j, o + j - 1, {1, 3, 1}[[t]], 2], {j, 1, u}] + Sum[If[t == 3, x, 1]*b[u + j - 1, o - j, 2, {1, 3, 1}[[h]]], {j, 1, o}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][ b[n, 0, 0, 0]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jun 07 2018, from Maple *) CROSSREFS Column k=0 gives A295974. Last elements of rows for n>3 give: A001250, A260786, 2*A000111. Row sums give A000142. Cf. A227884, A230695, A230797, A231384, A232933, A242783, A242819, A242820, A296054. Sequence in context: A058054 A054588 A084106 * A263691 A160657 A222087 Adjacent sequences:  A295984 A295985 A295986 * A295988 A295989 A295990 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Dec 01 2017 STATUS approved

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Last modified May 18 19:58 EDT 2021. Contains 344002 sequences. (Running on oeis4.)