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A156042 A(n,k) for n >= k in triangular ordering, where A(n,k) is the number of compositions (ordered partitions) of n into k parts, with the first part greater than or equal to all other parts.

%I A156042 #21 Nov 14 2014 12:12:19
%S A156042 1,1,2,1,2,4,1,3,6,11,1,3,8,17,32,1,4,11,26,54,102,1,4,13,35,82,172,
%T A156042 331,1,5,17,48,120,272,567,1101,1,5,20,63,170,412,918,1906,3724,1,6,
%U A156042 24,81,235,607,1434,3152,6518,12782,1,6,28,102,317,872,2180,5049,10978,22616,44444
%N A156042 A(n,k) for n >= k in triangular ordering, where A(n,k) is the number of compositions (ordered partitions) of n into k parts, with the first part greater than or equal to all other parts.
%C A156042 The value is smaller than the number of compositions (ordered partitions) of n into k parts and at least the number of (unordered) partitions.
%H A156042 Alois P. Heinz, <a href="/A156042/b156042.txt">Rows n = 1..100, flattened</a>
%e A156042 A(5,3) = 8 and the 8 compositions of 5 into 3 parts with first part maximal are:
%e A156042 [5,0,0], [4,1,0], [4,0,1], [3,2,0], [3,0,2], [3,1,1], [2,2,1], [2,1,2].
%e A156042 1
%e A156042 1  2
%e A156042 1  2  4
%e A156042 1  3  6  11
%e A156042 1  3  8  17  32
%e A156042 1  4  11 26  54  102
%p A156042 b:= proc(n,i,m) option remember;
%p A156042       if n<0 then 0
%p A156042     elif n=0 then 1
%p A156042     elif i=1 then `if`(n<=m, 1, 0)
%p A156042     else add(b(n-k, i-1, m), k=0..m)
%p A156042       fi
%p A156042     end:
%p A156042 A:= (n,k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n):
%p A156042 seq(seq(A(n,k), k=1..n), n=1..12); # _Alois P. Heinz_, Jun 14 2009
%t A156042 nn=10; Table[Table[Coefficient[Series[Sum[x^i((1-x^(i+1))/(1-x))^(k-1), {i, 0, n}], {x, 0, nn}], x^n], {k, 1, n}], {n, 1, nn}]//Grid (* _Geoffrey Critzer_, Jul 15 2013 *)
%Y A156042 A156041 is the whole of the square. A156043 is the diagonal. See also A156039 and A156040.
%K A156042 nonn,tabl
%O A156042 1,3
%A A156042 _Jack W Grahl_, Feb 02 2009
%E A156042 More terms from _Alois P. Heinz_, Jun 14 2009