A183561 Number of partitions of n containing a clique of size 4.

1, 0, 1, 1, 3, 3, 5, 6, 10, 13, 20, 23, 35, 44, 61, 78, 103, 131, 174, 219, 285, 355, 456, 567, 721, 894, 1117, 1382, 1718, 2109, 2607, 3180, 3902, 4747, 5789, 7010, 8500, 10251, 12373, 14867, 17868, 21369, 25584, 30505, 36372, 43233, 51350, 60834, 72039

Offset

4,5

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 = 4..5000

Formula

G.f.: (1-Product_{j>0} (1-x^(4*j)+x^(5*j))) / (Product_{j>0} (1-x^j)).

Example

a(10) = 5, because 5 partitions of 10 contain (at least) one clique of size 4: [1,1,1,1,2,2,2], [1,1,2,2,2,2], [1,1,1,1,3,3], [1,1,1,1,2,4], [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=4, [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=4..50);

Mathematica

max = 50; f = (1 - Product[1 - x^(4j) + x^(5j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x] , 4] (* Jean-François Alcover, Oct 01 2014 *)

Crossrefs

4th column of A183568. Cf. A000041, A183558, A183559, A183560, A183562, A183563, A183564, A183565, A183566, A183567.

Sequence in context: A241090 A091607 A276434 * A300183 A222704 A095950

Adjacent sequences: A183558 A183559 A183560 * A183562 A183563 A183564

Keyword

nonn

Author

Alois P. Heinz, Jan 05 2011

Status

approved