cp's OEIS Frontend

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.

%I A184957 #33 Dec 12 2024 11:08:49
%S A184957 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 A184957 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 A184957 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 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.
%C A184957 If the final diagonal is omitted, this gives the triangular array visible in A156041 and A186807.
%H A184957 Michael De Vlieger, <a href="/A184957/b184957.txt">Table of n, a(n) for n = 1..1275</a> (rows 1 <= n <= 50, flattened).
%H A184957 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 A184957 T(n,k) = A156041(n-k,k).
%F A184957 O.g.f.:  Sum_{i>=1} x^i/(1 - y*(x - x^(i+1))/(1-x)). - _Geoffrey Critzer_, Jul 15 2013
%e A184957 Triangle begins:
%e A184957   [1],
%e A184957   [1, 1],
%e A184957   [1, 1, 1],
%e A184957   [1, 2, 1, 1],
%e A184957   [1, 2, 3, 1, 1],
%e A184957   [1, 3, 4, 4, 1, 1],
%e A184957   [1, 3, 6, 7, 5, 1, 1],
%e A184957   [1, 4, 8, 11, 11, 6, 1, 1],
%e A184957   [1, 4, 11, 17, 19, 16, 7, 1, 1],
%e A184957   [1, 5, 13, 26, 32, 31, 22, 8, 1, 1],
%e A184957   [1, 5, 17, 35, 54, 56, 48, 29, 9, 1, 1],
%e A184957   ...
%p A184957 # The following Maple program is a modification of _Alois P. Heinz_'s program for A156041
%p A184957 b:= proc(n, i, m) option remember;
%p A184957        if n<0 then 0 elif n=0 then 1 elif i=1 then
%p A184957       `if`(n<=m, 1, 0) else add(b(n-k, i-1, m), k=0..m) fi
%p A184957     end:
%p A184957 A:= (n, k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n):
%p A184957 [seq([seq(A(d-k, k), k=1..d)], d=1..14)];
%t A184957 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 A184957 Cf. A156040, A156041, A186807, A079500 (row sums).
%K A184957 nonn,tabl
%O A184957 1,8
%A A184957 _N. J. A. Sloane_, Feb 27 2011