A184957 Triangle read by rows: T(n,k) (n >= 1, 1 <= k <= n) is the number of compositions of n into k parts the first of which is >= all the other parts.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 1, 3, 4, 4, 1, 1, 1, 3, 6, 7, 5, 1, 1, 1, 4, 8, 11, 11, 6, 1, 1, 1, 4, 11, 17, 19, 16, 7, 1, 1, 1, 5, 13, 26, 32, 31, 22, 8, 1, 1, 1, 5, 17, 35, 54, 56, 48, 29, 9, 1, 1, 1, 6, 20, 48, 82, 102, 93, 71, 37, 10, 1, 1, 1, 6, 24, 63, 120, 172, 180, 148, 101, 46, 11, 1, 1, 1, 7, 28, 81, 170, 272, 331, 302, 227, 139, 56, 12, 1, 1

Offset: 1

Author

N. J. A. Sloane, Feb 27 2011

Keywords

• nonn
• tabl

Comments

If the final diagonal is omitted, this gives the triangular array visible in A156041 and A186807.

Examples

			Triangle begins:
  [1],
  [1, 1],
  [1, 1, 1],
  [1, 2, 1, 1],
  [1, 2, 3, 1, 1],
  [1, 3, 4, 4, 1, 1],
  [1, 3, 6, 7, 5, 1, 1],
  [1, 4, 8, 11, 11, 6, 1, 1],
  [1, 4, 11, 17, 19, 16, 7, 1, 1],
  [1, 5, 13, 26, 32, 31, 22, 8, 1, 1],
  [1, 5, 17, 35, 54, 56, 48, 29, 9, 1, 1],
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows 1 <= n <= 50, flattened).
• Gal Gross, Maximally additively reducible subsets of the integers, arXiv:1908.05220 [math.CO], 2019.

Crossrefs

Cf. A156040, A156041, A186807, A079500 (row sums).

Programs

Maple

# The following Maple program is a modification of Alois P. Heinz's program for A156041
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(d-k, k), k=1..d)], d=1..14)];

Mathematica

Map[Select[#,#>0&]&,Drop[nn=11;CoefficientList[Series[Sum[x^i/(1-y(x-x^(i+1))/(1-x)),{i,1,nn}],{x,0,nn}],{x,y}],1]]//Grid (* Geoffrey Critzer, Jul 15 2013 *)

Formula

T(n,k) = A156041(n-k,k).

O.g.f.: Sum_{i>=1} x^i/(1 - y*(x - x^(i+1))/(1-x)). - Geoffrey Critzer, Jul 15 2013