A183560 Number of partitions of n containing a clique of size 3.

1, 0, 1, 2, 3, 3, 6, 8, 13, 15, 24, 30, 44, 54, 77, 98, 134, 165, 222, 279, 367, 454, 588, 731, 936, 1148, 1454, 1788, 2241, 2732, 3400, 4140, 5106, 6183, 7579, 9157, 11156, 13406, 16249, 19482, 23489, 28042, 33666, 40087, 47914, 56851

Offset

3,4

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 = 3..1000

Formula

G.f.: (1-Product_{j>0} (1-x^(3*j)+x^(4*j))) / (Product_{j>0} (1-x^j)).

Example

a(9) = 6, because 6 partitions of 9 contain (at least) one clique of size 3: [1,1,1,2,2,2], [2,2,2,3], [1,1,1,3,3], [3,3,3], [1,1,1,2,4], [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=3, [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=3..50);

Mathematica

max = 50; f = (1 - Product[1 - x^(3j) + x^(4j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 3] (* Jean-François Alcover, Oct 01 2014 *)

Crossrefs

Column k=3 of A183568.

Cf. A000041, A183558, A183559, A183561, A183562, A183563, A183564, A183565, A183566, A183567.

Sequence in context: A027100 A346030 A261090 * A060840 A357654 A276096

Adjacent sequences: A183557 A183558 A183559 * A183561 A183562 A183563

Keyword

nonn

Author

Alois P. Heinz, Jan 05 2011

Status

approved