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A242352 Number T(n,k) of isoscent sequences of length n with exactly k descents; triangle T(n,k), n>=0, 0<=k<=n+2-ceiling(2*sqrt(n+1)), read by rows. 13
1, 1, 2, 4, 1, 9, 6, 21, 29, 2, 51, 124, 28, 127, 499, 241, 10, 323, 1933, 1667, 216, 1, 835, 7307, 10142, 2765, 98, 2188, 27166, 56748, 27214, 2637, 22, 5798, 99841, 299485, 227847, 44051, 1546, 2, 15511, 363980, 1514445, 1708700, 563444, 46947, 570 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

An isoscent sequence of length n is an integer sequence [s(1),...,s(n)] with s(1) = 0 and 0 <= s(i) <= 1 plus the number of level steps in [s(1),...,s(i)].

Columns k=0-10 give: A001006, A243474, A243475, A243476, A243477, A243478, A243479, A243480, A243481, A243482, A243483.

Row sums give A000110.

Last elements of rows give A243484.

LINKS

Joerg Arndt and Alois P. Heinz, Rows n = 0..114, flattened

EXAMPLE

T(4,0) = 9: [0,0,0,0], [0,0,0,1], [0,0,0,2], [0,0,0,3], [0,0,1,1], [0,0,1,2], [0,0,2,2], [0,1,1,1], [0,1,1,2].

T(4,1) = 6: [0,0,1,0], [0,0,2,0], [0,0,2,1], [0,1,0,0], [0,1,0,1], [0,1,1,0].

T(5,2) = 2: [0,0,2,1,0], [0,1,0,1,0].

Triangle T(n,k) begins:

:    1;

:    1;

:    2;

:    4,     1;

:    9,     6;

:   21,    29,     2;

:   51,   124,    28;

:  127,   499,   241,    10;

:  323,  1933,  1667,   216,    1;

:  835,  7307, 10142,  2765,   98;

: 2188, 27166, 56748, 27214, 2637, 22;

MAPLE

b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(

      `if`(j<i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 0$2)):

seq(T(n), n=0..15);

MATHEMATICA

b[n_, i_, t_] := b[n, i, t] = If[n<1, 1, Expand[Sum[If[j<i, x, 1]*b[n-1, j, t + If[j == i, 1, 0]], {j, 0, t+1}]]]; T[n_] := Function[{p}, Table[ Coefficient[ p, x, i], {i, 0, Exponent[p, x]}]][b[n-1, 0, 0]]; Table[T[n], {n, 0, 15}] // Flatten (* Jean-Fran├žois Alcover, Feb 09 2015, after Maple *)

CROSSREFS

Cf. A048993 (for counting level steps), A242351 (for counting ascents), A137251 (ascent sequences counting ascents), A238858 (ascent sequences counting descents), A242153 (ascent sequences counting level steps), A083479.

Sequence in context: A091958 A116424 A135306 * A270953 A240717 A166900

Adjacent sequences:  A242349 A242350 A242351 * A242353 A242354 A242355

KEYWORD

nonn,tabf

AUTHOR

Joerg Arndt and Alois P. Heinz, May 11 2014

STATUS

approved

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Last modified July 20 01:38 EDT 2019. Contains 325168 sequences. (Running on oeis4.)