A183562 Number of partitions of n containing a clique of size 5.

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

Alois P. Heinz, Table of n, a(n) for n = 5..1000

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

5th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183563, A183564, A183565, A183566, A183567.

Sequence in context: A108962 A091608 A317081 * A222705 A241381 A237365

Adjacent sequences: A183559 A183560 A183561 * A183563 A183564 A183565

Keyword

nonn

Author

Alois P. Heinz, Jan 05 2011

Status

approved