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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.
5

%I #25 Nov 05 2019 00:59:32

%S 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,

%T 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,

%U 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

%N 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.

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

%H Michael De Vlieger, <a href="/A184957/b184957.txt">Table of n, a(n) for n = 1..1275</a> (rows 1 <= n <= 50, flattened).

%H Gal Gross, <a href="https://arxiv.org/abs/1908.05220">Maximally additively reducible subsets of the integers</a>, arXiv:1908.05220 [math.CO], 2019.

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

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

%e Triangle begins:

%e [1],

%e [1, 1],

%e [1, 1, 1],

%e [1, 2, 1, 1],

%e [1, 2, 3, 1, 1],

%e [1, 3, 4, 4, 1, 1],

%e [1, 3, 6, 7, 5, 1, 1],

%e [1, 4, 8, 11, 11, 6, 1, 1],

%e [1, 4, 11, 17, 19, 16, 7, 1, 1],

%e [1, 5, 13, 26, 32, 31, 22, 8, 1, 1],

%e [1, 5, 17, 35, 54, 56, 48, 29, 9, 1, 1],

%e ...

%p # The following Maple program is a modification of _Alois P. Heinz_'s program for A156041

%p b:= proc(n, i, m) option remember;

%p if n<0 then 0 elif n=0 then 1 elif i=1 then

%p `if`(n<=m, 1, 0) else add(b(n-k, i-1, m), k=0..m) fi

%p end:

%p A:= (n, k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n):

%p [seq([seq(A(d-k, k), k=1..d)], d=1..14)];

%t 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 *)

%Y Cf. A156040, A156041, A186807.

%K nonn,tabl

%O 1,8

%A _N. J. A. Sloane_, Feb 27 2011