cp's OEIS Frontend

A183564 Number of partitions of n containing a clique of size 7.

%I A183564 #17 Jun 12 2025 08:02:39
%S A183564 1,0,1,1,2,2,4,5,8,9,14,17,25,30,42,53,72,87,117,144,188,231,298,365,
%T A183564 466,567,714,871,1085,1316,1630,1972,2422,2918,3562,4280,5195,6219,
%U A183564 7507,8966,10773,12815,15335,18196,21680,25653,30453
%N A183564 Number of partitions of n containing a clique of size 7.
%C A183564 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 A183564 Alois P. Heinz, <a href="/A183564/b183564.txt">Table of n, a(n) for n = 7..1000</a>
%F A183564 G.f.: (1-Product_{j>0} (1-x^(7*j)+x^(8*j))) / (Product_{j>0} (1-x^j)).
%F A183564 a(n) = A000041(n) - A184642(n). - _Vaclav Kotesovec_, Jun 12 2025
%e A183564 a(13) = 4, because 4 partitions of 13 contain (at least) one clique of size 7: [1,1,1,1,1,1,1,2,2,2], [1,1,1,1,1,1,1,3,3], [1,1,1,1,1,1,1,2,4], [1,1,1,1,1,1,1,6].
%p A183564 b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
%p A183564       add((l->`if`(j=7, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
%p A183564     end:
%p A183564 a:= n-> (l-> l[2])(b(n, n)):
%p A183564 seq(a(n), n=7..55);
%t A183564 max = 55; f = (1 - Product[1 - x^(7j) + x^(8j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 7] (* _Jean-François Alcover_, Oct 01 2014 *)
%Y A183564 7th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183563, A183565, A183566, A183567.
%K A183564 nonn
%O A183564 7,5
%A A183564 _Alois P. Heinz_, Jan 05 2011