%I #33 May 04 2019 11:24:34

%S 1,1,1,1,1,4,1,11,3,1,26,25,1,57,128,17,1,120,525,229,2,1,247,1901,

%T 1819,172,1,502,6371,11172,3048,53,1,1013,20291,58847,33065,2751,7,1,

%U 2036,62407,280158,275641,56905,1422,1,4083,187272,1242859,1945529,771451,61966,436

%N 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.

%C 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)].

%C Columns k=0-10 give: A000012, A000295, A243228, A243229, A243230, A243231, A243232, A243233, A243234, A243235, A243236.

%C Row sums give A000110.

%C Last elements of rows give A243237.

%H Joerg Arndt and Alois P. Heinz, <a href="/A242351/b242351.txt">Rows n = 0..100, flattened</a>

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

%e 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].

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

%e Triangle T(n,k) begins:

%e 1;

%e 1;

%e 1, 1;

%e 1, 4;

%e 1, 11, 3;

%e 1, 26, 25;

%e 1, 57, 128, 17;

%e 1, 120, 525, 229, 2;

%e 1, 247, 1901, 1819, 172;

%e 1, 502, 6371, 11172, 3048, 53;

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

%e ...

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

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

%p end:

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

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

%t 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 *)

%Y 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.

%K nonn,tabf

%O 0,6

%A _Joerg Arndt_ and _Alois P. Heinz_, May 11 2014