OFFSET
1,5
COMMENTS
A(n,k) is of course smaller than the number of ordered partitions of n into k parts and at least the number of partitions into k parts in descending order.
The sums of the antidiagonals give A079500 - 1. - N. J. A. Sloane, Feb 26 2011
For an alternative definition of essentially the same sequence, as a triangle, and which avoids the use of parts of size zero, see A184957. - N. J. A. Sloane, Feb 27 2011
LINKS
R. H. Hardin, Table of n, a(n) for n = 1..2278
FORMULA
A(n,k)= [[x^n]]Sum_{i=0..n} x^i*((1 - x^(i+1))/(1-x))^(k-1). - Geoffrey Critzer, Jul 15 2013
EXAMPLE
The array A(n,k) begins:
1 1 1 1 1 1 1 1 1 ...
1 2 3 4 5 6 7 8 9 ...
1 2 4 7 11 16 22 29 ...
1 3 6 11 19 31 48 ...
1 3 8 17 32 56 ...
1 4 11 26 54 ...
1 4 13 35 ...
...
The antidiagonals are:
1,
1, 1,
1, 2, 1,
1, 2, 3, 1,
1, 3, 4, 4, 1,
1, 3, 6, 7, 5, 1,
1, 4, 8, 11, 11, 6, 1,
1, 4, 11, 17, 19, 16, 7, 1,
1, 5, 13, 26, 32, 31, 22, 8, 1,
...
A(3,5) = 11 and the 11 partition of 3 into 5 parts of this type are: (3,0,0,0,0), (2,1,0,0,0), (2,0,1,0,0), (2,0,0,1,0), (2,0,0,0,1), (1,1,1,0,0), (1,1,0,1,0), (1,1,0,0,1), (1,0,1,1,0), (1,0,1,0,1), (1,0,0,1,1).
MAPLE
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-1), d=1..14); # Alois P. Heinz, Jun 14 2009
MATHEMATICA
(* Returns rectangular array *) 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, nn}], {n, 1, nn}]//Grid (* Geoffrey Critzer, Jul 15 2013 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Jack W Grahl, Feb 02 2009, Feb 11 2009
EXTENSIONS
More terms from Alois P. Heinz, Jun 14 2009
Edited by N. J. A. Sloane, Feb 26 2011
STATUS
approved