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
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.
Programs
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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)];
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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
Comments