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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.
5
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, 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, 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
OFFSET
1,8
COMMENTS
If the final diagonal is omitted, this gives the triangular array visible in A156041 and A186807.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows 1 <= n <= 50, flattened).
Gal Gross, Maximally additively reducible subsets of the integers, arXiv:1908.05220 [math.CO], 2019.
FORMULA
T(n,k) = A156041(n-k,k).
E.g.f.: Sum_{i>=1} x^i/(1 - y*( x - x^(i+1))/(1-x) )/(1-x). - Geoffrey Critzer, Jul 15 2013
EXAMPLE
Triangle begins:
[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, 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, 48, 29, 9, 1, 1],
...
MAPLE
# The following Maple program is a modification of Alois P. Heinz's program for A156041
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)], d=1..14)];
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
N. J. A. Sloane, Feb 27 2011
STATUS
approved