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A183563 Number of partitions of n containing a clique of size 6. 13
1, 0, 1, 1, 2, 2, 5, 5, 8, 10, 15, 18, 27, 33, 47, 57, 78, 96, 129, 159, 208, 258, 330, 407, 517, 635, 798, 978, 1217, 1482, 1833, 2225, 2729, 3303, 4028, 4856, 5885, 7070, 8528, 10211, 12259, 14628, 17494, 20800, 24777, 29378, 34867 (list; graph; refs; listen; history; text; internal format)
OFFSET

6,5

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 = 6..1000

FORMULA

G.f.: (1-Product_{j>0} (1-x^(6*j)+x^(7*j))) / (Product_{j>0} (1-x^j)).

EXAMPLE

a(10) = 2, because 2 partitions of 10 contain (at least) one clique of size 6: [1,1,1,1,1,1,2,2], [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=6, [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=6..55);

MATHEMATICA

max = 55; f = (1 - Product[1 - x^(6j) + x^(7j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 6] (* Jean-Fran├žois Alcover, Oct 01 2014 *)

CROSSREFS

6th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183564, A183565, A183566, A183567.

Sequence in context: A073707 A238945 A091609 * A222706 A240495 A304393

Adjacent sequences:  A183560 A183561 A183562 * A183564 A183565 A183566

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jan 05 2011

STATUS

approved

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Last modified February 21 22:10 EST 2020. Contains 332113 sequences. (Running on oeis4.)