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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
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)].
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-François 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: A116424 A372883 A135306 * A270953 A240717 A166900
KEYWORD
nonn,tabf
AUTHOR
Joerg Arndt and Alois P. Heinz, May 11 2014
STATUS
approved