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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.

%I A242351 #33 May 04 2019 11:24:34
%S A242351 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 A242351 1819,172,1,502,6371,11172,3048,53,1,1013,20291,58847,33065,2751,7,1,
%U A242351 2036,62407,280158,275641,56905,1422,1,4083,187272,1242859,1945529,771451,61966,436
%N 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.
%C A242351 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 A242351 Columns k=0-10 give: A000012, A000295, A243228, A243229, A243230, A243231, A243232, A243233, A243234, A243235, A243236.
%C A242351 Row sums give A000110.
%C A242351 Last elements of rows give A243237.
%H A242351 Joerg Arndt and Alois P. Heinz, <a href="/A242351/b242351.txt">Rows n = 0..100, flattened</a>
%e A242351 T(4,0) = 1: [0,0,0,0].
%e A242351 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 A242351 T(4,2) = 3: [0,0,1,2], [0,1,0,1], [0,1,1,2].
%e A242351 Triangle T(n,k) begins:
%e A242351   1;
%e A242351   1;
%e A242351   1,    1;
%e A242351   1,    4;
%e A242351   1,   11,     3;
%e A242351   1,   26,    25;
%e A242351   1,   57,   128,    17;
%e A242351   1,  120,   525,   229,     2;
%e A242351   1,  247,  1901,  1819,   172;
%e A242351   1,  502,  6371, 11172,  3048,   53;
%e A242351   1, 1013, 20291, 58847, 33065, 2751, 7;
%e A242351   ...
%p A242351 b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
%p A242351       `if`(j>i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
%p A242351     end:
%p A242351 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 0$2)):
%p A242351 seq(T(n), n=0..15);
%t A242351 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 A242351 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 A242351 nonn,tabf
%O A242351 0,6
%A A242351 _Joerg Arndt_ and _Alois P. Heinz_, May 11 2014