cp's OEIS Frontend

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

%I A183567 #17 Jun 12 2025 08:04:33
%S A183567 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 A183567 381,462,580,703,874,1055,1303,1569,1921,2309,2808,3363,4070,4859,
%U A183567 5848,6964,8342,9903,11817,13988,16623,19626,23240,27363,32297
%N A183567 Number of partitions of n containing a clique of size 10.
%C A183567 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 A183567 Alois P. Heinz, <a href="/A183567/b183567.txt">Table of n, a(n) for n = 10..1000</a>
%F A183567 G.f.: (1-Product_{j>0} (1-x^(10*j)+x^(11*j))) / (Product_{j>0} (1-x^j)).
%F A183567 a(n) = A000041(n) - A184645(n). - _Vaclav Kotesovec_, Jun 12 2025
%e A183567 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 A183567 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A183567       add((l->`if`(j=10, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A183567     end:
%p A183567 a:= n-> (l-> l[2])(b(n, n)):
%p A183567 seq(a(n), n=10..60);
%t A183567 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-François Alcover_, Oct 01 2014 *)
%t A183567 Table[Length[Select[IntegerPartitions[n],MemberQ[Length/@Split[#],10]&]],{n,10,60}] (* _Harvey P. Dale_, Oct 02 2021 *)
%Y A183567 10th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566.
%K A183567 nonn
%O A183567 10,5
%A A183567 _Alois P. Heinz_, Jan 05 2011