A183564 Number of partitions of n containing a clique of size 7.

1, 0, 1, 1, 2, 2, 4, 5, 8, 9, 14, 17, 25, 30, 42, 53, 72, 87, 117, 144, 188, 231, 298, 365, 466, 567, 714, 871, 1085, 1316, 1630, 1972, 2422, 2918, 3562, 4280, 5195, 6219, 7507, 8966, 10773, 12815, 15335, 18196, 21680, 25653, 30453

Offset

7,5

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 = 7..1000

Formula

G.f.: (1-Product_{j>0} (1-x^(7*j)+x^(8*j))) / (Product_{j>0} (1-x^j)).

Example

a(13) = 4, because 4 partitions of 13 contain (at least) one clique of size 7: [1,1,1,1,1,1,1,2,2,2], [1,1,1,1,1,1,1,3,3], [1,1,1,1,1,1,1,2,4], [1,1,1,1,1,1,1,6].

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=7, [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=7..55);

Mathematica

max = 55; f = (1 - Product[1 - x^(7j) + x^(8j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 7] (* Jean-François Alcover, Oct 01 2014 *)

Crossrefs

7th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183565, A183566, A183567.

Sequence in context: A326445 A069906 A304332 * A222707 A326525 A326630

Adjacent sequences: A183561 A183562 A183563 * A183565 A183566 A183567

Keyword

nonn

Author

Alois P. Heinz, Jan 05 2011

Status

approved