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A183560 Number of partitions of n containing a clique of size 3. 12
1, 0, 1, 2, 3, 3, 6, 8, 13, 15, 24, 30, 44, 54, 77, 98, 134, 165, 222, 279, 367, 454, 588, 731, 936, 1148, 1454, 1788, 2241, 2732, 3400, 4140, 5106, 6183, 7579, 9157, 11156, 13406, 16249, 19482, 23489, 28042, 33666, 40087, 47914, 56851 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,4

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 = 3..1000

FORMULA

G.f.: (1-Product_{j>0} (1-x^(3*j)+x^(4*j))) / (Product_{j>0} (1-x^j)).

EXAMPLE

a(9) = 6, because 6 partitions of 9 contain (at least) one clique of size 3: [1,1,1,2,2,2], [2,2,2,3], [1,1,1,3,3], [3,3,3], [1,1,1,2,4], [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=3, [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=3..50);

MATHEMATICA

max = 50; f = (1 - Product[1 - x^(3j) + x^(4j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 3] (* Jean-Fran├žois Alcover, Oct 01 2014 *)

CROSSREFS

Column k=3 of A183568.

Cf. A000041, A183558, A183559, A183561, A183562, A183563, A183564, A183565, A183566, A183567.

Sequence in context: A187505 A027100 A261090 * A060840 A276096 A074717

Adjacent sequences:  A183557 A183558 A183559 * A183561 A183562 A183563

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jan 05 2011

STATUS

approved

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Last modified February 29 05:25 EST 2020. Contains 332353 sequences. (Running on oeis4.)