A156042 A(n,k) for n >= k in triangular ordering, where A(n,k) is the number of compositions (ordered partitions) of n into k parts, with the first part greater than or equal to all other parts.

1, 1, 2, 1, 2, 4, 1, 3, 6, 11, 1, 3, 8, 17, 32, 1, 4, 11, 26, 54, 102, 1, 4, 13, 35, 82, 172, 331, 1, 5, 17, 48, 120, 272, 567, 1101, 1, 5, 20, 63, 170, 412, 918, 1906, 3724, 1, 6, 24, 81, 235, 607, 1434, 3152, 6518, 12782, 1, 6, 28, 102, 317, 872, 2180, 5049, 10978, 22616, 44444

Offset

1,3

Comments

The value is smaller than the number of compositions (ordered partitions) of n into k parts and at least the number of (unordered) partitions.

Links

Alois P. Heinz, Rows n = 1..100, flattened

Example

A(5,3) = 8 and the 8 compositions of 5 into 3 parts with first part maximal are:
[5,0,0], [4,1,0], [4,0,1], [3,2,0], [3,0,2], [3,1,1], [2,2,1], [2,1,2].
1
1  2
1  2  4
1  3  6  11
1  3  8  17  32
1  4  11 26  54  102

Maple

b:= proc(n, i, m) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i=1 then `if`(n<=m, 1, 0)
    else add(b(n-k, i-1, m), k=0..m)
      fi
    end:
A:= (n, k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n):
seq(seq(A(n, k), k=1..n), n=1..12); # Alois P. Heinz, Jun 14 2009

Mathematica

nn=10; Table[Table[Coefficient[Series[Sum[x^i((1-x^(i+1))/(1-x))^(k-1), {i, 0, n}], {x, 0, nn}], x^n], {k, 1, n}], {n, 1, nn}]//Grid (* Geoffrey Critzer, Jul 15 2013 *)

Crossrefs

A156041 is the whole of the square. A156043 is the diagonal. See also A156039 and A156040.

Sequence in context: A306915 A270743 A209750 * A287691 A227926 A357121

Adjacent sequences: A156039 A156040 A156041 * A156043 A156044 A156045

Keyword

nonn,tabl

Author

Jack W Grahl, Feb 02 2009

Extensions

More terms from Alois P. Heinz, Jun 14 2009

Status

approved