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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.

%I A242352 #34 Feb 09 2015 10:33:32
%S A242352 1,1,2,4,1,9,6,21,29,2,51,124,28,127,499,241,10,323,1933,1667,216,1,
%T A242352 835,7307,10142,2765,98,2188,27166,56748,27214,2637,22,5798,99841,
%U A242352 299485,227847,44051,1546,2,15511,363980,1514445,1708700,563444,46947,570
%N 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.
%C A242352 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 A242352 Columns k=0-10 give: A001006, A243474, A243475, A243476, A243477, A243478, A243479, A243480, A243481, A243482, A243483.
%C A242352 Row sums give A000110.
%C A242352 Last elements of rows give A243484.
%H A242352 Joerg Arndt and Alois P. Heinz, <a href="/A242352/b242352.txt">Rows n = 0..114, flattened</a>
%e A242352 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].
%e A242352 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].
%e A242352 T(5,2) = 2: [0,0,2,1,0], [0,1,0,1,0].
%e A242352 Triangle T(n,k) begins:
%e A242352 :    1;
%e A242352 :    1;
%e A242352 :    2;
%e A242352 :    4,     1;
%e A242352 :    9,     6;
%e A242352 :   21,    29,     2;
%e A242352 :   51,   124,    28;
%e A242352 :  127,   499,   241,    10;
%e A242352 :  323,  1933,  1667,   216,    1;
%e A242352 :  835,  7307, 10142,  2765,   98;
%e A242352 : 2188, 27166, 56748, 27214, 2637, 22;
%p A242352 b:= proc(n, i, t) option remember; `if`(n<1, 1, expand(add(
%p A242352       `if`(j<i, x, 1) *b(n-1, j, t+`if`(j=i, 1, 0)), j=0..t+1)))
%p A242352     end:
%p A242352 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 0$2)):
%p A242352 seq(T(n), n=0..15);
%t A242352 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 A242352 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.
%K A242352 nonn,tabf
%O A242352 0,3
%A A242352 _Joerg Arndt_ and _Alois P. Heinz_, May 11 2014