A183566 Number of partitions of n containing a clique of size 9.

1, 0, 1, 1, 2, 2, 4, 4, 7, 9, 13, 15, 23, 27, 38, 47, 63, 77, 103, 126, 165, 201, 258, 315, 401, 487, 611, 743, 924, 1118, 1382, 1664, 2041, 2455, 2989, 3583, 4340, 5185, 6248, 7446, 8930, 10604, 12668, 15002, 17848, 21083, 24987, 29435, 34776, 40860

Offset

9,5

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 = 9..1000

Formula

G.f.: (1-Product_{j>0} (1-x^(9*j)+x^(10*j))) / (Product_{j>0} (1-x^j)).

Example

a(12) = 1, because 1 partition of 12 contains (at least) one clique of size 9: [1,1,1,1,1,1,1,1,1,3].

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=9, [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=9..60);

Mathematica

max=60; f=(1-Product[1-x^(9j)+x^(10j), {j, 1, max}])/Product[1-x^j, {j, 1, max}]; s=Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 9] (* Jean-François Alcover, Oct 01 2014 *)

Crossrefs

9th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183567.

Sequence in context: A055636 A206559 A143419 * A222709 A034396 A364193

Adjacent sequences: A183563 A183564 A183565 * A183567 A183568 A183569

Keyword

nonn

Author

Alois P. Heinz, Jan 05 2011

Status

approved