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A183567 Number of partitions of n containing a clique of size 10. 12

%I

%S 1,0,1,1,2,2,4,4,7,8,13,15,22,26,37,45,61,74,99,120,157,192,247,299,

%T 381,462,580,703,874,1055,1303,1569,1921,2309,2808,3363,4070,4859,

%U 5848,6964,8342,9903,11817,13988,16623,19626,23240,27363,32297

%N Number of partitions of n containing a clique of size 10.

%C 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.

%H Alois P. Heinz, <a href="/A183567/b183567.txt">Table of n, a(n) for n = 10..1000</a>

%F G.f.: (1-Product_{j>0} (1-x^(10*j)+x^(11*j))) / (Product_{j>0} (1-x^j)).

%e a(14) = 2, because 2 partitions of 14 contain (at least) one clique of size 10: [1,1,1,1,1,1,1,1,1,1,2,2], [1,1,1,1,1,1,1,1,1,1,4].

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

%p add((l->`if`(j=10, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))

%p end:

%p a:= n-> (l-> l[2])(b(n, n)):

%p seq(a(n), n=10..60);

%t max = 60; f = (1 - Product[1 - x^(10j) + x^(11j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 10] (* _Jean-Fran├žois Alcover_, Oct 01 2014 *)

%Y 10th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566.

%K nonn

%O 10,5

%A _Alois P. Heinz_, Jan 05 2011

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Last modified February 21 03:02 EST 2020. Contains 332086 sequences. (Running on oeis4.)