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A242351 Number T(n,k) of isoscent sequences of length n with exactly k ascents; triangle T(n,k), n>=0, 0<=k<=n+3-ceiling(2*sqrt(n+2)), read by rows. 12
1, 1, 1, 1, 1, 4, 1, 11, 3, 1, 26, 25, 1, 57, 128, 17, 1, 120, 525, 229, 2, 1, 247, 1901, 1819, 172, 1, 502, 6371, 11172, 3048, 53, 1, 1013, 20291, 58847, 33065, 2751, 7, 1, 2036, 62407, 280158, 275641, 56905, 1422, 1, 4083, 187272, 1242859, 1945529, 771451, 61966, 436 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

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: A000012, A000295, A243228, A243229, A243230, A243231, A243232, A243233, A243234, A243235, A243236.

Row sums give A000110.

Last elements of rows give A243237.

LINKS

Joerg Arndt and Alois P. Heinz, Rows n = 0..100, flattened

EXAMPLE

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

T(4,1) = 11: [0,0,0,1], [0,0,0,2], [0,0,0,3], [0,0,1,0], [0,0,1,1], [0,0,2,0], [0,0,2,1], [0,0,2,2], [0,1,0,0], [0,1,1,0], [0,1,1,1].

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

Triangle T(n,k) begins:

  1;

  1;

  1,    1;

  1,    4;

  1,   11,     3;

  1,   26,    25;

  1,   57,   128,    17;

  1,  120,   525,   229,     2;

  1,  247,  1901,  1819,   172;

  1,  502,  6371, 11172,  3048,   53;

  1, 1013, 20291, 58847, 33065, 2751, 7;

  ...

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), A242352 (for counting descents), A137251 (ascent sequences counting ascents), A238858 (ascent sequences counting descents), A242153 (ascent sequences counting level steps), A083479.

Sequence in context: A091156 A092288 A111964 * A124324 A178519 A094503

Adjacent sequences:  A242348 A242349 A242350 * A242352 A242353 A242354

KEYWORD

nonn,tabf

AUTHOR

Joerg Arndt and Alois P. Heinz, May 11 2014

STATUS

approved

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Last modified June 16 04:54 EDT 2019. Contains 324145 sequences. (Running on oeis4.)