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

1,8

Comments

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

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.

Formula

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

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

Example

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],
...

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 *)

Crossrefs

Cf. A156040, A156041, A186807.

Sequence in context: A187660 A066170 A046854 * A340811 A340812 A228349

Adjacent sequences: A184954 A184955 A184956 * A184958 A184959 A184960

Keyword

nonn,tabl

Author

N. J. A. Sloane, Feb 27 2011

Status

approved