%I #21 Nov 14 2014 12:12:19
%S 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 331,1,5,17,48,120,272,567,1101,1,5,20,63,170,412,918,1906,3724,1,6,
%U 24,81,235,607,1434,3152,6518,12782,1,6,28,102,317,872,2180,5049,10978,22616,44444
%N 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 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 Alois P. Heinz, <a href="/A156042/b156042.txt">Rows n = 1..100, flattened</a>
%e A(5,3) = 8 and the 8 compositions of 5 into 3 parts with first part maximal are:
%e [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 1
%e 1 2
%e 1 2 4
%e 1 3 6 11
%e 1 3 8 17 32
%e 1 4 11 26 54 102
%p b:= proc(n,i,m) option remember;
%p if n<0 then 0
%p elif n=0 then 1
%p elif i=1 then `if`(n<=m, 1, 0)
%p else add(b(n-k, i-1, m), k=0..m)
%p fi
%p end:
%p A:= (n,k)-> add(b(n-m, k-1, m), m=ceil(n/k)..n):
%p seq(seq(A(n,k), k=1..n), n=1..12); # _Alois P. Heinz_, Jun 14 2009
%t 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 A156041 is the whole of the square. A156043 is the diagonal. See also A156039 and A156040.
%K nonn,tabl
%O 1,3
%A _Jack W Grahl_, Feb 02 2009
%E More terms from _Alois P. Heinz_, Jun 14 2009