A183565 Number of partitions of n containing a clique of size 8.

1, 0, 1, 1, 2, 2, 4, 4, 8, 9, 13, 16, 24, 28, 40, 49, 66, 82, 110, 132, 175, 214, 274, 336, 428, 520, 655, 798, 990, 1203, 1486, 1793, 2200, 2653, 3227, 3880, 4701, 5622, 6779, 8092, 9701, 11546, 13793, 16355, 19466, 23029, 27290, 32199, 38048, 44752, 52719

Offset

8,5

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 = 8..1000

Formula

G.f.: (1-Product_{j>0} (1-x^(8*j)+x^(9*j))) / (Product_{j>0} (1-x^j)).

Example

a(12) = 2, because 2 partitions of 12 contain (at least) one clique of size 8: [1,1,1,1,1,1,1,1,2,2], [1,1,1,1,1,1,1,1,4].

Maple

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=8, [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=8..60);

Mathematica

max = 60; f = (1 - Product[1 - x^(8j) + x^(9j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 8] (* Jean-François Alcover, Oct 01 2014 *)
c8[n_]:=If[MemberQ[Tally[n][[All, 2]], 8], 1, 0]; Table[Total[c8/@ IntegerPartitions[ x]], {x, 8, 60}] (* Harvey P. Dale, Aug 12 2018 *)

Crossrefs

8th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183566, A183567.

Sequence in context: A217208 A287136 A351788 * A222708 A324843 A306692

Adjacent sequences: A183562 A183563 A183564 * A183566 A183567 A183568

Keyword

nonn

Author

Alois P. Heinz, Jan 05 2011

Status

approved