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A183559 Number of partitions of n containing a clique of size 2. 12
1, 0, 2, 2, 3, 5, 9, 10, 16, 23, 31, 43, 60, 75, 106, 140, 179, 237, 310, 389, 508, 647, 815, 1032, 1305, 1617, 2033, 2527, 3117, 3857, 4764, 5812, 7142, 8711, 10585, 12866, 15605, 18803, 22716, 27325, 32774, 39286, 47016, 56019, 66819, 79456, 94273, 111766 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,3

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 = 2..1000

FORMULA

G.f.: (1-Product_{j>0} (1-x^(2*j)+x^(3*j))) / (Product_{j>0} (1-x^j)).

EXAMPLE

a(7) = 5, because 5 partitions of 7 contain (at least) one clique of size 2: [1,1,1,2,2], [1,1,2,3], [2,2,3], [1,3,3], [1,1,5].

MAPLE

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],

      add((l->`if`(j=2, [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=2..50);

MATHEMATICA

max = 50; f = (1 - Product[1 - x^(2j) + x^(3j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 2] (* Jean-Fran├žois Alcover, Oct 01 2014 *)

CROSSREFS

Column k=2 of A183568.

Cf. A000041, A183558, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.

Sequence in context: A039822 A025591 A028409 * A080553 A226956 A141602

Adjacent sequences:  A183556 A183557 A183558 * A183560 A183561 A183562

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jan 05 2011

STATUS

approved

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Last modified February 17 18:14 EST 2020. Contains 332005 sequences. (Running on oeis4.)