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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. 5
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, 331, 1, 5, 17, 48, 120, 272, 567, 1101, 1, 5, 20, 63, 170, 412, 918, 1906, 3724, 1, 6, 24, 81, 235, 607, 1434, 3152, 6518, 12782, 1, 6, 28, 102, 317, 872, 2180, 5049, 10978, 22616, 44444 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
The value is smaller than the number of compositions (ordered partitions) of n into k parts and at least the number of (unordered) partitions.
LINKS
EXAMPLE
A(5,3) = 8 and the 8 compositions of 5 into 3 parts with first part maximal are:
[5,0,0], [4,1,0], [4,0,1], [3,2,0], [3,0,2], [3,1,1], [2,2,1], [2,1,2].
1
1 2
1 2 4
1 3 6 11
1 3 8 17 32
1 4 11 26 54 102
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(n, k), k=1..n), n=1..12); # Alois P. Heinz, Jun 14 2009
MATHEMATICA
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 *)
CROSSREFS
A156041 is the whole of the square. A156043 is the diagonal. See also A156039 and A156040.
Sequence in context: A306915 A270743 A209750 * A287691 A227926 A357121
KEYWORD
nonn,tabl
AUTHOR
Jack W Grahl, Feb 02 2009
EXTENSIONS
More terms from Alois P. Heinz, Jun 14 2009
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)