A183568 -id:A183568 - cp's OEIS frontend

A007690 Number of partitions of n in which no part occurs just once.

1, 0, 1, 1, 2, 1, 4, 2, 6, 5, 9, 7, 16, 11, 22, 20, 33, 28, 51, 42, 71, 66, 100, 92, 147, 131, 199, 193, 275, 263, 385, 364, 516, 511, 694, 686, 946, 925, 1246, 1260, 1650, 1663, 2194, 2202, 2857, 2928, 3721, 3813, 4866, 4967, 6257, 6487, 8051, 8342, 10369

Offset: 0

N. J. A. Sloane, Robert G. Wilson v

nonn

Also number of partitions of n into parts, each larger than 1, such that consecutive integers do not both appear as parts. Example: a(6)=4 because we have [6], [4,2], [3,3] and [2,2,2]. - Emeric Deutsch, Feb 16 2006
Also number of partitions of n into parts divisible by 2 or 3. - Alexander E. Holroyd (holroyd(AT)math.ubc.ca), May 28 2008
Infinite convolution product of [1,0,1,1,1,1,1] aerated n-1 times. i.e. [1,0,1,1,1,1,1] * [1,0,0,0,1,0,1] * [1,0,0,0,0,0,1] * ... . - Mats Granvik, Aug 07 2009

			a(6) = 4 because we have [3,3], [2,2,2], [2,2,1,1] and [1,1,1,1,1,1].
G.f. = 1 + x^2 + x^3 + 2*x^4 + x^5 + 4*x^6 + 2*x^7 + 6*x^8 + 5*x^9 + 9*x^10 + ...
G.f. = q + q^49 + q^73 + 2*q^97 + q^121 + 4*q^145 + 2*q^169 + 6*q^193 + ...

G. E. Andrews, Number Theory, Dover Publications, 1994. page 197. MR1298627
G. E. Andrews, The Theory of Partitions, Addison-Wesley, Reading, Mass., 1976, p. 14, Example 9.
I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, (2.5.6).
R. Honsberger, Mathematical Gems III, M.A.A., 1985, p. 242.
P. A. MacMahon, Combinatory Analysis, Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 54, Article 300.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
A. E. Holroyd, T. M. Liggett and D. Romik, Integrals, partitions and cellular automata, arXiv:math/0302216 [math.PR], 2003.
Eric Weisstein's World of Mathematics, Partition Function P

Cf. A000041, A055922, A055923, A114917, A114918, A183558, A183568.

Cf. A100405, A160974-A160990.

G:= mul((1-x^j+x^(2*j))/(1-x^j), j=1..70): Gser:=series(G, x, 60): seq(coeff(Gser, x, n), n=0..54); # Emeric Deutsch, Feb 10 2006

nn=40;CoefficientList[Series[Product[1/(1-x^i)-x^i,{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Dec 02 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^6] / (QPochhammer[ x^2] QPochhammer[ x^3]), {x, 0, n}]; (* Michael Somos, Feb 22 2015 *)
nmax = 60; CoefficientList[Series[Product[(1 + x^(3*k))/(1 - x^(2*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 23 2015 *)
Table[Length@Select[Tally /@ IntegerPartitions@n, AllTrue[#, Last[#] > 1 &] &], {n, 0, 54}] (* Robert Price, Aug 17 2020 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A)), n))}; /* Michael Somos, Apr 21 2004 */

G.f.: Product_{k>0 is a multiple of 2 or 3} (1/(1-x^k)). - Christian G. Bower, Jun 23 2000
G.f.: Product_{j>=1} (1+x^(3*j)) / (1-x^(2*j)). - Jon Perry, Mar 29 2004
Euler transform of period 6 sequence [0, 1, 1, 1, 0, 1, ...]. - Michael Somos, Apr 21 2004
G.f. is a period 1 Fourier series which satisfies f(-1 / (864 t)) = 1/6 (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A137566. - Michael Somos, Jan 26 2008
From Alois P. Heinz, Oct 09 2011: (Start)
a(n) = A000041(n) - A183558(n).
a(n) = A183568(n,0) - A183568(n,1).
G.f.: Product_{j>0} (1-x^j+x^(2*j)) / (1-x^j). (End)
a(n) ~ exp(2*Pi*sqrt(n)/3)/(6*sqrt(2)*n). - Vaclav Kotesovec, Sep 23 2015
a(n) = A000009(n/3) - Sum_{k>=1} (-1)^k a(n - k*(3*k +/- 1)). - Peter J. Taylor, May 16 2019

Minor edits by Vaclav Kotesovec, Aug 23 2015

A183558 Number of partitions of n containing a clique of size 1.

Original entry on oeis.org

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

Alois P. Heinz, Jan 05 2011

nonn

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.

			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)

Alois P. Heinz, Table of n, a(n) for n = 0..5000

Column k=1 of A183568.
Cf. A000041, A007690, A183559, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.
Cf. A052485, A090858, A117571, A127002, A325241, A325242, A325244.

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);

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 *)

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)

a(0)=0 prepended by Gus Wiseman, Apr 19 2019

A243978 Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n where the minimal multiplicity of any part is k.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1, 0, 6, 0, 0, 0, 1, 0, 7, 2, 1, 0, 0, 1, 0, 13, 1, 0, 0, 0, 0, 1, 0, 16, 4, 0, 1, 0, 0, 0, 1, 0, 25, 2, 2, 0, 0, 0, 0, 0, 1, 0, 33, 6, 1, 0, 1, 0, 0, 0, 0, 1, 0, 49, 4, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 61, 9, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1, 0, 90, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 113, 16, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 156, 9, 7, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 0

Joerg Arndt and Alois P. Heinz, Jun 28 2014

T(0,0) = 1 by convention.
Columns k=0-10 give: A000007, A183558, A244515, A244516, A244517, A244518, A245037, A245038, A245039, A245040, A245041.
Row sums are A000041.

			Triangle starts:
00:  1;
01:  0,   1;
02:  0,   1,  1;
03:  0,   2,  0, 1;
04:  0,   3,  1, 0, 1;
05:  0,   6,  0, 0, 0, 1;
06:  0,   7,  2, 1, 0, 0, 1;
07:  0,  13,  1, 0, 0, 0, 0, 1;
08:  0,  16,  4, 0, 1, 0, 0, 0, 1;
09:  0,  25,  2, 2, 0, 0, 0, 0, 0, 1;
10:  0,  33,  6, 1, 0, 1, 0, 0, 0, 0, 1;
11:  0,  49,  4, 2, 0, 0, 0, 0, 0, 0, 0, 1;
12:  0,  61,  9, 3, 2, 0, 1, 0, 0, 0, 0, 0, 1;
13:  0,  90,  6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
14:  0, 113, 16, 2, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1;
15:  0, 156,  9, 7, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
16:  0, 198, 23, 3, 4, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1;
17:  0, 269, 18, 5, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
18:  0, 334, 34, 9, 3, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 1;
19:  0, 448, 27, 8, 3, 2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
20:  0, 556, 51, 7, 6, 3, 2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
...
The A000041(9) = 30 partitions of 9 with the least multiplicities of any part are:
01:  [ 1 1 1 1 1 1 1 1 1 ]   9
02:  [ 1 1 1 1 1 1 1 2 ]   1
03:  [ 1 1 1 1 1 1 3 ]   1
04:  [ 1 1 1 1 1 2 2 ]   2
05:  [ 1 1 1 1 1 4 ]   1
06:  [ 1 1 1 1 2 3 ]   1
07:  [ 1 1 1 1 5 ]   1
08:  [ 1 1 1 2 2 2 ]   3
09:  [ 1 1 1 2 4 ]   1
10:  [ 1 1 1 3 3 ]   2
11:  [ 1 1 1 6 ]   1
12:  [ 1 1 2 2 3 ]   1
13:  [ 1 1 2 5 ]   1
14:  [ 1 1 3 4 ]   1
15:  [ 1 1 7 ]   1
16:  [ 1 2 2 2 2 ]   1
17:  [ 1 2 2 4 ]   1
18:  [ 1 2 3 3 ]   1
19:  [ 1 2 6 ]   1
20:  [ 1 3 5 ]   1
21:  [ 1 4 4 ]   1
22:  [ 1 8 ]   1
23:  [ 2 2 2 3 ]   1
24:  [ 2 2 5 ]   1
25:  [ 2 3 4 ]   1
26:  [ 2 7 ]   1
27:  [ 3 3 3 ]   3
28:  [ 3 6 ]   1
29:  [ 4 5 ]   1
30:  [ 9 ]   1
Therefore row n=9 is [0, 25, 2, 2, 0, 0, 0, 0, 0, 1].

Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..10010 (rows 0..140, flattened)

Cf. A183568, A242451 (the same for compositions).

Cf. A091602 (partitions by max multiplicity of any part).

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1, k) +add(b(n-i*j, i-1, k), j=max(1, k)..n/i)))
    end:
T:= (n, k)-> b(n$2, k) -`if`(n=0 and k=0, 0, b(n$2, k+1)):
seq(seq(T(n, k), k=0..n), n=0..14);

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, b[n, i-1, k] + Sum[b[n-i*j, i-1, k], {j, Max[1, k], n/i}]]]; T[n_, k_] := b[n, n, k] - If[n == 0 && k == 0, 0, b[n, n, k+1]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 08 2015, translated from Maple *)

A116645 Number of partitions of n having no doubletons. By a doubleton in a partition we mean an occurrence of a part exactly twice (the partition [4,(3,3),2,2,2,(1,1)] of 18 has two doubletons, shown between parentheses).

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 8, 10, 13, 20, 26, 33, 46, 58, 75, 101, 125, 157, 206, 253, 317, 403, 494, 608, 760, 926, 1131, 1393, 1685, 2038, 2487, 2985, 3585, 4331, 5168, 6172, 7392, 8771, 10410, 12382, 14622, 17258, 20400, 23975, 28159, 33115, 38739, 45298, 53000

Offset: 0

Emeric Deutsch and Vladeta Jovovic, Feb 20 2006

nonn

Number of partitions of n having no part that appears exactly twice.

Infinite convolution product of [1,1,0,1,1,1,1,1,1,1] aerated n-1 times. I.e., [1,1,0,1,1,1,1,1,1,1] * [1,0,1,0,0,0,1,0,1,0] * [1,0,0,1,0,0,0,0,0,1] * ... - Mats Granvik, Gary W. Adamson, Aug 07 2009

			a(4) = 3 because we have [4],[3,1] and [1,1,1,1] (the partitions [2,2] and [2,1,1] do not qualify since each of them has a doubleton).

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Column 0 of A116644.

Cf. A000041, A007690, A116595, A183559, A183568.

h:=product((1-x^(2*j)+x^(3*j))/(1-x^j),j=1..60): hser:=series(h,x=0,60): seq(coeff(hser,x,n),n=0..56);

nn=48;CoefficientList[Series[Product[1/(1-x^i)-x^(2i),{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Sep 30 2013 *)

G.f.: Product_{j>=1} (1-x^(2j)+x^(3j))/(1-x^j).
G.f. for the number of partitions of n having no part that appears exactly m times is Product_{k>0} (1/(1-x^k)-x^(m*k)).
a(n) = A000041(n) - A183559(n) = A183568(n,0) - A183568(n,2). - Alois P. Heinz, Oct 09 2011
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-2*x) + exp(-4*x)) dx = 0.64673501839556449802623523266221107725058748270577037891948... - Vaclav Kotesovec, Jun 12 2025

A183563 Number of partitions of n containing a clique of size 6.

Original entry on oeis.org

1, 0, 1, 1, 2, 2, 5, 5, 8, 10, 15, 18, 27, 33, 47, 57, 78, 96, 129, 159, 208, 258, 330, 407, 517, 635, 798, 978, 1217, 1482, 1833, 2225, 2729, 3303, 4028, 4856, 5885, 7070, 8528, 10211, 12259, 14628, 17494, 20800, 24777, 29378, 34867

Offset: 6

Alois P. Heinz, Jan 05 2011

nonn

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.

			a(10) = 2, because 2 partitions of 10 contain (at least) one clique of size 6: [1,1,1,1,1,1,2,2], [1,1,1,1,1,1,4].

Alois P. Heinz, Table of n, a(n) for n = 6..1000

6th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183562, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=6, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=6..55);

max = 55; f = (1 - Product[1 - x^(6j) + x^(7j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 6] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(6*j)+x^(7*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A184641(n). - Vaclav Kotesovec, Jun 12 2025

A118807 Number of partitions of n having no parts with multiplicity 3.

Original entry on oeis.org

1, 1, 2, 2, 5, 6, 9, 12, 19, 24, 34, 43, 62, 77, 105, 132, 177, 220, 287, 356, 462, 570, 723, 888, 1121, 1370, 1705, 2074, 2570, 3111, 3816, 4601, 5617, 6743, 8170, 9777, 11794, 14058, 16858, 20029, 23932, 28334, 33692, 39772, 47133, 55468, 65471, 76840

Offset: 0

Emeric Deutsch, Apr 29 2006

nonn

Column 0 of A118806.

Infinite convolution product of [1,1,1,0,1,1,1,1,1,1] aerated n-1 times. I.e., [1,1,1,0,1,1,1,1,1,1] * [1,0,1,0,1,0,0,0,1,0] * [1,0,0,1,0,0,1,0,0,0] * ... - Mats Granvik, Gary W. Adamson, Aug 07 2009

			a(6) = 9 because among the 11 (=A000041(6)) partitions of 6 only [2,2,2] and [3,1,1,1] have parts with multiplicity 3.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A000041, A118806, A007690, A116645, A183560, A183568.

g:=product(1+x^j+x^(2*j)+x^(4*j)/(1-x^j),j=1..60): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=0..50);

nmax = 50; CoefficientList[Series[Product[(1 - x^(3*k) + x^(4*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Mar 07 2016 *)

G.f.: Product_{j>=1} (1 + x^j + x^(2j) + x^(4j)/(1-x^j)).
a(n) = A000041(n) - A183560(n) = A183568(n,0) - A183568(n,3). - Alois P. Heinz, Oct 09 2011
a(n) ~ exp(sqrt((Pi^2/3 + 4*r)*n)) * sqrt(Pi^2/6 + 2*r) / (4*Pi*n), where r = Integral_{x=0..oo} log(1 + exp(-x) - exp(-3*x) + exp(-5*x)) dx = 0.73597677748514060768682570953508781551028221145343244320009... - Vaclav Kotesovec, Jun 12 2025

A183559 Number of partitions of n containing a clique of size 2.

Original entry on oeis.org

1, 0, 2, 2, 3, 5, 9, 10, 16, 23, 31, 43, 60, 75, 106, 140, 179, 237, 310, 389, 508, 647, 815, 1032, 1305, 1617, 2033, 2527, 3117, 3857, 4764, 5812, 7142, 8711, 10585, 12866, 15605, 18803, 22716, 27325, 32774, 39286, 47016, 56019, 66819, 79456, 94273, 111766

Offset: 2

Alois P. Heinz, Jan 05 2011

nonn

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.

			a(7) = 5, because 5 partitions of 7 contain (at least) one clique of size 2: [1,1,1,2,2], [1,1,2,3], [2,2,3], [1,3,3], [1,1,5].

Alois P. Heinz, Table of n, a(n) for n = 2..1000

Column k=2 of A183568.

Cf. A000041, A183558, A183560, A183561, A183562, A183563, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=2, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=2..50);

max = 50; f = (1 - Product[1 - x^(2j) + x^(3j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 2] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(2*j)+x^(3*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A116645(n). - Vaclav Kotesovec, Jun 12 2025

A183560 Number of partitions of n containing a clique of size 3.

Original entry on oeis.org

1, 0, 1, 2, 3, 3, 6, 8, 13, 15, 24, 30, 44, 54, 77, 98, 134, 165, 222, 279, 367, 454, 588, 731, 936, 1148, 1454, 1788, 2241, 2732, 3400, 4140, 5106, 6183, 7579, 9157, 11156, 13406, 16249, 19482, 23489, 28042, 33666, 40087, 47914, 56851

Offset: 3

Alois P. Heinz, Jan 05 2011

nonn

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.

			a(9) = 6, because 6 partitions of 9 contain (at least) one clique of size 3: [1,1,1,2,2,2], [2,2,2,3], [1,1,1,3,3], [3,3,3], [1,1,1,2,4], [1,1,1,6].

Alois P. Heinz, Table of n, a(n) for n = 3..1000

Column k=3 of A183568.

Cf. A000041, A183558, A183559, A183561, A183562, A183563, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=3, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=3..50);

max = 50; f = (1 - Product[1 - x^(3j) + x^(4j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 3] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(3*j)+x^(4*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A118807(n). - Vaclav Kotesovec, Jun 12 2025

A183561 Number of partitions of n containing a clique of size 4.

Original entry on oeis.org

1, 0, 1, 1, 3, 3, 5, 6, 10, 13, 20, 23, 35, 44, 61, 78, 103, 131, 174, 219, 285, 355, 456, 567, 721, 894, 1117, 1382, 1718, 2109, 2607, 3180, 3902, 4747, 5789, 7010, 8500, 10251, 12373, 14867, 17868, 21369, 25584, 30505, 36372, 43233, 51350, 60834, 72039

Offset: 4

Alois P. Heinz, Jan 05 2011

nonn

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.

			a(10) = 5, because 5 partitions of 10 contain (at least) one clique of size 4: [1,1,1,1,2,2,2], [1,1,2,2,2,2], [1,1,1,1,3,3], [1,1,1,1,2,4], [1,1,1,1,6].

Alois P. Heinz, Table of n, a(n) for n = 4..5000

4th column of A183568. Cf. A000041, A183558, A183559, A183560, A183562, A183563, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=4, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=4..50);

max = 50; f = (1 - Product[1 - x^(4j) + x^(5j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x] , 4] (* Jean-François Alcover, Oct 01 2014 *)

G.f.: (1-Product_{j>0} (1-x^(4*j)+x^(5*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A184639(n). - Vaclav Kotesovec, Jun 12 2025

A183562 Number of partitions of n containing a clique of size 5.

Original entry on oeis.org

1, 0, 1, 1, 2, 3, 5, 5, 9, 11, 16, 21, 31, 36, 52, 65, 88, 110, 148, 180, 238, 295, 379, 469, 600, 731, 926, 1133, 1413, 1725, 2141, 2590, 3194, 3864, 4719, 5692, 6924, 8301, 10049, 12026, 14468, 17263, 20694, 24586, 29359, 34804, 41372

Offset: 5

Alois P. Heinz, Jan 05 2011

nonn

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.

			a(11) = 5, because 5 partitions of 11 contain (at least) one clique of size 5: [1,1,1,1,1,2,2,2], [1,2,2,2,2,2], [1,1,1,1,1,3,3], [1,1,1,1,1,2,4], [1,1,1,1,1,6].

Alois P. Heinz, Table of n, a(n) for n = 5..1000

5th column of A183568. Cf. A000041, A183558, A183559, A183560, A183561, A183563, A183564, A183565, A183566, A183567.

b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, [0, 0],
      add((l->`if`(j=5, [l[1]$2], l))(b(n-i*j, i-1)), j=0..n/i)))
    end:
a:= n-> (l-> l[2])(b(n, n)):
seq(a(n), n=5..55);

max = 55; f = (1 - Product[1 - x^(5j) + x^(6j), {j, 1, max}])/Product[1 - x^j, {j, 1, max}]; s = Series[f, {x, 0, max}]; Drop[CoefficientList[s, x], 5] (* Jean-François Alcover, Oct 01 2014 *)
Table[Count[IntegerPartitions[n,{5,PartitionsP[n]}],?(MemberQ[ Length/@ Split[ #],5]&)],{n,5,60}] (* _Harvey P. Dale, Feb 02 2019 *)

G.f.: (1-Product_{j>0} (1-x^(5*j)+x^(6*j))) / (Product_{j>0} (1-x^j)).

a(n) = A000041(n) - A184640(n). - Vaclav Kotesovec, Jun 12 2025

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