login
A183558
Number of partitions of n containing a clique of size 1.
29
0, 1, 1, 2, 3, 6, 7, 13, 16, 25, 33, 49, 61, 90, 113, 156, 198, 269, 334, 448, 556, 726, 902, 1163, 1428, 1827, 2237, 2817, 3443, 4302, 5219, 6478, 7833, 9632, 11616, 14197, 17031, 20712, 24769, 29925, 35688, 42920, 50980, 61059, 72318, 86206, 101837, 120941
OFFSET
0,4
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
FORMULA
G.f.: (1-Product_{j>0} (1-x^(j)+x^(2*j))) / (Product_{j>0} (1-x^j)).
From Vaclav Kotesovec, Nov 15 2016: (Start)
a(n) = A000041(n) - A007690(n).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)*n). (End)
EXAMPLE
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].
From Gus Wiseman, Apr 19 2019: (Start)
The a(1) = 1 through a(8) = 16 partitions are the following. The Heinz numbers of these partitions are given by A052485 (weak numbers).
(1) (2) (3) (4) (5) (6) (7) (8)
(21) (31) (32) (42) (43) (53)
(211) (41) (51) (52) (62)
(221) (321) (61) (71)
(311) (411) (322) (332)
(2111) (3111) (331) (422)
(21111) (421) (431)
(511) (521)
(2221) (611)
(3211) (3221)
(4111) (4211)
(31111) (5111)
(211111) (32111)
(41111)
(311111)
(2111111)
(End)
MAPLE
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
add((l->`if`(j=1, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
end:
a:= n-> b(n$2)[2]:
seq(a(n), n=0..50);
MATHEMATICA
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 *)
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jan 05 2011
EXTENSIONS
a(0)=0 prepended by Gus Wiseman, Apr 19 2019
STATUS
approved