A238122 Irregular triangle read by rows: T(n,k) gives the number of ballot sequences of length n having k descents, n>=0, 0<=k<=A083920(n-1).

1, 1, 2, 3, 1, 5, 5, 7, 16, 3, 11, 43, 21, 1, 15, 99, 101, 17, 22, 215, 373, 145, 9, 30, 430, 1174, 836, 146, 4, 42, 834, 3337, 3846, 1324, 112, 1, 56, 1529, 8642, 15002, 8786, 1615, 66, 77, 2765, 21148, 52132, 47013, 15403, 1582, 32, 101, 4792, 48713, 164576, 214997, 112106, 21895, 1310, 14

Offset

0,3

Comments

Same as A238121, with zeros omitted.

Columns k=0-10 give: A000041, A241794, A241795, A241796, A241797, A241798, A241799, A241800, A241801, A241802, A241803.

T(2n,n) gives A241804.

T(2n+1,n) gives A241805.

Row sums are A000085.

T(n*(n+1)/2,n*(n-1)/2) = 1.

Links

Joerg Arndt and Alois P. Heinz, Rows n = 0..50, flattened

Example

T(5,0) = 7: [1,1,1,1,1], [1,1,1,1,2], [1,1,1,2,2], [1,1,1,2,3], [1,1,2,2,3], [1,1,2,3,4], [1,2,3,4,5].
T(5,1) = 16: [1,1,1,2,1], [1,1,2,1,1], [1,1,2,1,2], [1,1,2,1,3], [1,1,2,2,1], [1,1,2,3,1], [1,1,2,3,2], [1,2,1,1,1], [1,2,1,1,2], [1,2,1,1,3], [1,2,1,2,3], [1,2,1,3,4], [1,2,3,1,1], [1,2,3,1,2], [1,2,3,1,4], [1,2,3,4,1].
T(5,2) = 3: [1,2,1,2,1], [1,2,1,3,1], [1,2,1,3,2].
Triangle starts:
00:   1;
01:   1;
02:   2;
03:   3,    1;
04:   5,    5;
05:   7,   16,      3;
06:  11,   43,     21,      1;
07:  15,   99,    101,     17;
08:  22,  215,    373,    145,      9;
09:  30,  430,   1174,    836,    146,      4;
10:  42,  834,   3337,   3846,   1324,    112,      1;
11:  56, 1529,   8642,  15002,   8786,   1615,     66;
12:  77, 2765,  21148,  52132,  47013,  15403,   1582,    32;
13: 101, 4792,  48713, 164576, 214997, 112106,  21895,  1310,  14;
14: 135, 8216, 108147, 484609, 874413, 672015, 215849, 26159, 932, 5;
...

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->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n-1, 1, [1])):
seq(T(n), n=0..14);

Mathematica

b[n_, v_, l_List] := b[n, v, l] = If[n<1, 1, Expand[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_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n-1, 1, {1}]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 11 2015, after Maple *)

Crossrefs

Sequence in context: A294223 A355618 A376414 * A214059 A195508 A049274

Adjacent sequences: A238119 A238120 A238121 * A238123 A238124 A238125

Keyword

nonn,tabf

Author

Joerg Arndt and Alois P. Heinz, Feb 21 2014

Status

approved