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A183567
Number of partitions of n containing a clique of size 10.
12
1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 13, 15, 22, 26, 37, 45, 61, 74, 99, 120, 157, 192, 247, 299, 381, 462, 580, 703, 874, 1055, 1303, 1569, 1921, 2309, 2808, 3363, 4070, 4859, 5848, 6964, 8342, 9903, 11817, 13988, 16623, 19626, 23240, 27363, 32297
OFFSET
10,5
COMMENTS
All parts of a number partition with the same value form a clique. The size of a clique is the number of elements in the clique.
LINKS
FORMULA
G.f.: (1-Product_{j>0} (1-x^(10*j)+x^(11*j))) / (Product_{j>0} (1-x^j)).
EXAMPLE
a(14) = 2, because 2 partitions of 14 contain (at least) one clique of size 10: [1,1,1,1,1,1,1,1,1,1,2,2], [1,1,1,1,1,1,1,1,1,1,4].
MAPLE
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->`if`(j=10, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=10..60);
MATHEMATICA
max = 60; f = (1 - Product[1 - x^(10j) + x^(11j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 10] (* Jean-François Alcover, Oct 01 2014 *)
Table[Length[Select[IntegerPartitions[n], MemberQ[Length/@Split[#], 10]&]], {n, 10, 60}] (* Harvey P. Dale, Oct 02 2021 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jan 05 2011
STATUS
approved