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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
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)].
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-François 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
KEYWORD
nonn,tabf
AUTHOR
Joerg Arndt and Alois P. Heinz, May 11 2014
STATUS
approved