A183559 Number of partitions of n containing a clique of size 2.

1, 0, 2, 2, 3, 5, 9, 10, 16, 23, 31, 43, 60, 75, 106, 140, 179, 237, 310, 389, 508, 647, 815, 1032, 1305, 1617, 2033, 2527, 3117, 3857, 4764, 5812, 7142, 8711, 10585, 12866, 15605, 18803, 22716, 27325, 32774, 39286, 47016, 56019, 66819, 79456, 94273, 111766

Offset

2,3

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 = 2..1000

Formula

G.f.: (1-Product_{j>0} (1-x^(2*j)+x^(3*j))) / (Product_{j>0} (1-x^j)).

Example

a(7) = 5, because 5 partitions of 7 contain (at least) one clique of size 2: [1,1,1,2,2], [1,1,2,3], [2,2,3], [1,3,3], [1,1,5].

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=2, [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=2..50);

Mathematica

max = 50; f = (1 - Product[1 - x^(2j) + x^(3j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 2] (* Jean-François Alcover, Oct 01 2014 *)

Crossrefs

Column k=2 of A183568.

Cf. A000041, A183558, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.

Sequence in context: A025591 A028409 A348850 * A080553 A350432 A226956

Adjacent sequences: A183556 A183557 A183558 * A183560 A183561 A183562

Keyword

nonn

Author

Alois P. Heinz, Jan 05 2011

Status

approved