login
Number of partitions of n containing a clique of size 1.
29

%I #38 Apr 26 2019 17:38:07

%S 0,1,1,2,3,6,7,13,16,25,33,49,61,90,113,156,198,269,334,448,556,726,

%T 902,1163,1428,1827,2237,2817,3443,4302,5219,6478,7833,9632,11616,

%U 14197,17031,20712,24769,29925,35688,42920,50980,61059,72318,86206,101837,120941

%N Number of partitions of n containing a clique of size 1.

%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="/A183558/b183558.txt">Table of n, a(n) for n = 0..5000</a>

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

%F From _Vaclav Kotesovec_, Nov 15 2016: (Start)

%F a(n) = A000041(n) - A007690(n).

%F a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). (End)

%e a(5) = 6, because 6 partitions of 5 contain (at least) one clique of size 1: [1,1,1,2], [1,2,2], [1,1,3], [2,3], [1,4], [5].

%e From _Gus Wiseman_, Apr 19 2019: (Start)

%e The a(1) = 1 through a(8) = 16 partitions are the following. The Heinz numbers of these partitions are given by A052485 (weak numbers).

%e (1) (2) (3) (4) (5) (6) (7) (8)

%e (21) (31) (32) (42) (43) (53)

%e (211) (41) (51) (52) (62)

%e (221) (321) (61) (71)

%e (311) (411) (322) (332)

%e (2111) (3111) (331) (422)

%e (21111) (421) (431)

%e (511) (521)

%e (2221) (611)

%e (3211) (3221)

%e (4111) (4211)

%e (31111) (5111)

%e (211111) (32111)

%e (41111)

%e (311111)

%e (2111111)

%e (End)

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

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

%p end:

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

%p seq(a(n), n=0..50);

%t max = 50; f = (1 - Product[1 - x^j + x^(2*j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; CoefficientList[s, x] (* _Jean-François Alcover_, Oct 01 2014. Edited by _Gus Wiseman_, Apr 19 2019 *)

%Y Column k=1 of A183568.

%Y Cf. A000041, A007690, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.

%Y Cf. A052485, A090858, A117571, A127002, A325241, A325242, A325244.

%K nonn

%O 0,4

%A _Alois P. Heinz_, Jan 05 2011

%E a(0)=0 prepended by _Gus Wiseman_, Apr 19 2019