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A183562
Number of partitions of n containing a clique of size 5.
12
1, 0, 1, 1, 2, 3, 5, 5, 9, 11, 16, 21, 31, 36, 52, 65, 88, 110, 148, 180, 238, 295, 379, 469, 600, 731, 926, 1133, 1413, 1725, 2141, 2590, 3194, 3864, 4719, 5692, 6924, 8301, 10049, 12026, 14468, 17263, 20694, 24586, 29359, 34804, 41372
OFFSET
5,5
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^(5*j)+x^(6*j))) / (Product_{j>0} (1-x^j)).
EXAMPLE
a(11) = 5, because 5 partitions of 11 contain (at least) one clique of size 5: [1,1,1,1,1,2,2,2], [1,2,2,2,2,2], [1,1,1,1,1,3,3], [1,1,1,1,1,2,4], [1,1,1,1,1,6].
MAPLE
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->`if`(j=5, [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=5..55);
MATHEMATICA
max = 55; f = (1 - Product[1 - x^(5j) + x^(6j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 5] (* Jean-François Alcover, Oct 01 2014 *)
Table[Count[IntegerPartitions[n, {5, PartitionsP[n]}], _?(MemberQ[ Length/@ Split[ #], 5]&)], {n, 5, 60}] (* Harvey P. Dale, Feb 02 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jan 05 2011
STATUS
approved