A238125 Triangle read by rows: T(n,k) gives the number of ballot sequences of length n having exactly k flat steps, n>=0, 0<=k<=n.

1, 1, 0, 1, 1, 0, 2, 1, 1, 0, 4, 3, 2, 1, 0, 9, 8, 6, 2, 1, 0, 22, 24, 17, 9, 3, 1, 0, 59, 70, 57, 29, 13, 3, 1, 0, 170, 224, 191, 108, 49, 17, 4, 1, 0, 516, 744, 663, 399, 201, 69, 23, 4, 1, 0, 1658, 2588, 2415, 1573, 802, 322, 104, 28, 5, 1, 0, 5583, 9317, 9108, 6249, 3343, 1408, 510, 137, 35, 5, 1, 0

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Feb 21 2014

Keywords

• nonn
• tabl

Comments

Also number of standard Young tableaux with n cells and exactly k successions. A succession is a pair of cells (v, v+1) lying in the same row.

Columns k=0-10 give: A237770, A238126, A238127, A241774, A241775, A241776, A241777, A241778, A241779, A241780, A241781.

T(2n,n) gives A241785.

Row sums are A000085.

Examples

			Triangle starts:
00:     1;
01:     1,     0;
02:     1,     1,     0;
03:     2,     1,     1,     0;
04:     4,     3,     2,     1,     0;
05:     9,     8,     6,     2,     1,    0;
06:    22,    24,    17,     9,     3,    1,    0;
07:    59,    70,    57,    29,    13,    3,    1,   0;
08:   170,   224,   191,   108,    49,   17,    4,   1,   0;
09:   516,   744,   663,   399,   201,   69,   23,   4,   1,  0;
10:  1658,  2588,  2415,  1573,   802,  322,  104,  28,   5,  1, 0;
11:  5583,  9317,  9108,  6249,  3343, 1408,  510, 137,  35,  5, 1, 0;
12: 19683, 34924, 35695, 25642, 14368, 6440, 2411, 751, 189, 42, 6, 1, 0;
...
		

Links

• Joerg Arndt and Alois P. Heinz, Rows n = 0..45, flattened

Programs

Maple

b:= proc(n, v, l) option remember; `if`(n<1, 1, expand(
      add(`if`(i=1 or l[i-1]>l[i], `if`(i=v, x, 1)*
      b(n-1, i, subsop(i=l[i]+1, l)), 0), i=1..nops(l))+
      b(n-1, nops(l)+1, [l[], 1])))
    end:
T:= n-> seq(coeff(b(n-1, 1, [1]), x, i), i=0..n):
seq(T(n), n=0..12);

Mathematica

b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Sum[If[i == 1 || l[[i-1]] > l[[i]], If[i == v, x, 1]*b[n-1, i, ReplacePart[l, i -> l[[i]]+1]], 0], {i, 1, Length[l]}] + b[n-1, Length[l]+1, Append[l, 1]]]; T[n_] := Table[Coefficient[b[n-1, 1, {1}], x, i], {i, 0, n}]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Jan 07 2015, translated from Maple *)