A079122 -id:A079122 - cp's OEIS frontend

A209816 Number of partitions of 2n in which every part is

1, 3, 7, 15, 30, 58, 105, 186, 318, 530, 863, 1380, 2164, 3345, 5096, 7665, 11395, 16765, 24418, 35251, 50460, 71669, 101050, 141510, 196888, 272293, 374423, 512081, 696760, 943442, 1271527, 1706159, 2279700, 3033772, 4021695, 5311627, 6990367, 9168321

Offset: 1

Clark Kimberling, Mar 13 2012

nonn

Also, the number of partitions of 3n in which n is the maximal part.
Also, the number of partitions of 3n into n parts. - Seiichi Manyama, May 07 2018
Also the number of multigraphical partitions of 2n, i.e., integer partitions that comprise the multiset of vertex-degrees of some multigraph. - Gus Wiseman, Oct 24 2018
Also number of partitions of 2n with at most n parts. Conjugate partitions map one to one to partitions of 2*n with each part <= n. - Wolfdieter Lang, May 21 2019

			The 7 partitions of 6 with parts <4 are as follows:
3+3, 3+2+1, 3+1+1+1
2+2+2, 2+2+1+1, 2+1+1+1+1
1+1+1+1+1+1.
Matching partitions of 2 into rationals as described:
1 + 1
1 + 3/3 + 1/3
1 + 1/3 + 1/3 + 1/3
2/3 + 2/3 + 2/3
2/3 + 2/3 + 1/3 + 1/3
2/3 + 1/3 + 1/3 + 1/3 + 1/3
1/3 + 1/3 + 1/3 + 1/3 + 1/3 + 1/3.
From _Seiichi Manyama_, May 07 2018: (Start)
n | Partitions of 3n into n parts
--+-------------------------------------------------
1 | 3;
2 | 5+1, 4+2, 3+3;
3 | 7+1+1, 6+2+1, 5+3+1, 5+2+2, 4+4+1, 4+3+2, 3+3+3; (End)
From _Gus Wiseman_, Oct 24 2018: (Start)
The a(1) = 1 through a(4) = 15 partitions:
  (11)  (22)    (33)      (44)
        (211)   (222)     (332)
        (1111)  (321)     (422)
                (2211)    (431)
                (3111)    (2222)
                (21111)   (3221)
                (111111)  (3311)
                          (4211)
                          (22211)
                          (32111)
                          (41111)
                          (221111)
                          (311111)
                          (2111111)
                          (11111111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Gus Wiseman, Multigraphs realizing each of the a(4) = 15 multigraphical partitions of 8.
Gus Wiseman, Multigraphs realizing each of the a(5) = 30 multigraphical partitions of 10.

Cf. A000041, A000070, A000569, A008284, A025065, A079122, A096373, A147878, A209815, A320911, A320921, A320924.

a209816 n = p [1..n] (2*n) where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Nov 14 2013

b:= proc(n, i) option remember;
      `if`(n=0, 1, `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(2*n, n):
seq(a(n), n=1..50);  # Alois P. Heinz, Jul 09 2012

f[n_] := Length[Select[IntegerPartitions[2 n], First[#] <= n &]]; Table[f[n], {n, 1, 30}] (* A209816 *)
Table[SeriesCoefficient[Product[1/(1-x^k),{k,1,n}],{x,0,2*n}],{n,1,20}] (* Vaclav Kotesovec, May 25 2015 *)
Table[Length@IntegerPartitions[3n, {n}], {n, 25}] (* Vladimir Reshetnikov, Jul 24 2016 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[2*n, n]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Aug 29 2016, after Alois P. Heinz *)

a(n) = A000041(2*n)-A000070(n-1). - Matthew Vandermast, Jul 16 2012

a(n) = Sum_{k=1..n} A008284(2*n, k) = A000041(2*n) - A000070(n-1), for n >= 1. - Wolfdieter Lang, May 21 2019

More terms from Alois P. Heinz, Jul 09 2012

A035294 Number of ways to partition 2n into distinct positive integers.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 15, 22, 32, 46, 64, 89, 122, 165, 222, 296, 390, 512, 668, 864, 1113, 1426, 1816, 2304, 2910, 3658, 4582, 5718, 7108, 8808, 10880, 13394, 16444, 20132, 24576, 29927, 36352, 44046, 53250, 64234, 77312, 92864, 111322, 133184, 159046

Offset: 0

N. J. A. Sloane, Bill Gosper

Also, number of partitions of 2n into odd numbers. - Vladeta Jovovic, Aug 17 2004
This sequence was originally defined as the expansion of sum ( q^n / product( 1-q^k, k=1..2*n), n=0..inf ). The present definition is due to Reinhard Zumkeller. Michael Somos points out that the equivalence of the two definitions follows from Andrews, page 19.
Also, number of partitions of 2n with max descent 1 and last part 1. - Wouter Meeussen, Mar 31 2013

			a(4)=6 [8=7+1=6+2=5+3=5+2+1=4+3+1=2*4].
G.f. = 1 + x + 2*x^2 + 4*x^3 + 6*x^4 + 10*x^5 + 15*x^6 + 22*x^7 + 46*x^9 + ...
G.f. = q + q^49 + 2*q^97 + 4*q^145 + 6*q^193 + 10*q^241 + 15*q^289 + ...

G. E. Andrews, The Theory of Partitions, Cambridge University Press, 1998, p. 19.

Alois P. Heinz, Table of n, a(n) for n = 0..10000
N. J. A. Sloane, Transforms
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000009, A000041, A058686, A262987, A282893.
Cf. A078408, A078406, A078407.
Cf. A079122, A079126, A079124, A079125, A067953.
Cf. A005408.

import Data.MemoCombinators (memo2, integral)
a035294 n = a035294_list !! n
a035294_list = f 1 where
   f x = (p' 1 (x - 1)) : f (x + 2)
   p' = memo2 integral integral p
   p _ 0 = 1
   p k m = if m < k then 0 else p' k (m - k) + p' (k + 2) m
-- Reinhard Zumkeller, Nov 27 2015

b:= proc(n, i) option remember; `if`(n=0, 1,
     `if`(i<1, 0, b(n, i-2)+`if`(i>n, 0, b(n-i, i))))
    end:
a:= n-> b(2*n, 2*n-1):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 11 2015

Table[Count[IntegerPartitions[2 n], q_ /; Union[q] == Sort[q]], {n, 16}];
Table[Count[IntegerPartitions[2 n], q_ /; Count[q, _?EvenQ] == 0], {n, 16}];
Table[Count[IntegerPartitions[2 n], q_ /; Last[q] == 1 && Max[q - PadRight[Rest[q], Length[q]]] <= 1 ], {n, 16}];
(* Wouter Meeussen, Mar 31 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] /QPochhammer[ x], {x, 0, 2 n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ x^8] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, May 06 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^(8*k+1)) * (1+x^(8*k+2))^2 * (1+x^(8*k+3))^2 * (1+x^(8*k+4))^3 * (1+x^(8*k+5))^2 * (1+x^(8*k+6))^2 * (1+x^(8*k+7)) * (1+x^(8*k+8))^3, {k,0,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 06 2015 *)
b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-2] + If[i>n, 0, b[n-i, i]]]]; a[n_] := b[2n, 2n-1]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Aug 30 2016, after Alois P. Heinz *)

{a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( eta(x^2 + A) / eta(x + A), n))};/* Michael Somos, Nov 01 2005 */

a(n) = A000009(2*n). - Michael Somos, Mar 03 2003
Expansion of Sum_{n >= 0} q^n / Product_{k = 1..2*n} (1 - q^k).
a(n) = T(2*n, 0), T as defined in A026835.
G.f.: Product_{i >= 0} ((1 + x^(8*i + 1)) * (1 + x^(8*i + 2))^2 * (1 + x^(8*i + 3))^2 * (1 + x^(8*i + 4))^3 * (1 + x^(8*i + 5))^2 * (1 + x^(8*i + 6))^2 * (1 + x^(8*i + 7)) * (1 + x^(8*i + 8))^3). - Vladeta Jovovic, Oct 10 2004
G.f.: (Sum_{k>=0} x^A074378(k)) / (Product_{k>0} (1 - x^k)) = f( x^3, x^5) / f(-x, -x^2) where f(, ) is Ramanujan's general theta function. - Michael Somos, Nov 01 2005
Euler transform of period 16 sequence [1, 1, 2, 1, 2, 0, 1, 0, 1, 0, 2, 1, 2, 1, 1, 0, ...]. - Michael Somos, Dec 17 2002
a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(11/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Oct 06 2015
a(n) = A000041(n) + A282893(n). - Michael Somos, Feb 24 2017
Convolution with A000041 is A058696. - Michael Somos, Feb 24 2017
Convolution with A097451 is A262987. - Michael Somos, Feb 24 2017
G.f.: 1/(1 - x)*Sum_{n>=0} x^floor((3*n+1)/2)/Product_{k = 1..n} (1 - x^k). - Peter Bala, Feb 04 2021
G.f.: Product_{n >= 1} (1 - q^(8*n))*(1 + q^(8*n-3))*(1 + q^(8*n-5))/(1 - q^n). - Peter Bala, Dec 30 2024

A182616 Number of partitions of 2n that contain odd parts.

Original entry on oeis.org

0, 1, 3, 8, 17, 35, 66, 120, 209, 355, 585, 946, 1498, 2335, 3583, 5428, 8118, 12013, 17592, 25525, 36711, 52382, 74173, 104303, 145698, 202268, 279153, 383145, 523105, 710655, 960863, 1293314, 1733281, 2313377, 3075425, 4073085, 5374806, 7067863, 9263076

Offset: 0

Omar E. Pol, Dec 03 2010

nonn

Bisection (even part) of A086543.

			For n=3 the partitions of 2n are
6 ....................... does not contains odd parts
3 + 3 ................... contains odd parts ........... *
4 + 2 ................... does not contains odd parts
2 + 2 + 2 ............... does not contains odd parts
5 + 1 ................... contains odd parts ........... *
3 + 2 + 1 ............... contains odd parts ........... *
4 + 1 + 1 ............... contains odd parts ........... *
2 + 2 + 1 + 1 ........... contains odd parts ........... *
3 + 1 + 1 + 1 ........... contains odd parts ........... *
2 + 1 + 1 + 1 + 1 ....... contains odd parts ........... *
1 + 1 + 1 + 1 + 1 + 1 ... contains odd parts ........... *
There are 8 partitions of 2n that contain odd parts.
Also p(2n)-p(n) = p(6)-p(3) = 11-3 = 8, where p(n) is the number of partitions of n, so a(3)=8.
From _Gus Wiseman_, Oct 18 2023: (Start)
For n > 0, also the number of integer partitions of 2n that do not contain n, ranked by A366321. For example, the a(1) = 1 through a(4) = 17 partitions are:
  (2)  (4)     (6)       (8)
       (31)    (42)      (53)
       (1111)  (51)      (62)
               (222)     (71)
               (411)     (332)
               (2211)    (521)
               (21111)   (611)
               (111111)  (2222)
                         (3221)
                         (3311)
                         (5111)
                         (22211)
                         (32111)
                         (221111)
                         (311111)
                         (2111111)
                         (11111111)
(End)

Cf. A304710.
Bisection of A086543, with ranks A366322.
The case of all odd parts is A035294, bisection of A000009.
The strict case is A365828.
These partitions have ranks A366530.
A000041 counts integer partitions, strict A000009.
A006477 counts partitions with at least one odd and even part, ranks A366532.
A047967 counts partitions with at least one even part, ranks A324929.
A086543 counts partitions of n not containing n/2, ranks A366319.
A366527 counts partitions of 2n with an even part, ranks A366529.
Cf. A001522, A006827, A058695, A078408, A079122, A231429, A365543, A366321.

with(combinat): a:= n-> numbpart(2*n) -numbpart(n): seq(a(n), n=0..35);

Table[Length[Select[IntegerPartitions[2n],n>0&&FreeQ[#,n]&]],{n,0,15}] (* Gus Wiseman, Oct 11 2023 *)
Table[Length[Select[IntegerPartitions[2n],Or@@OddQ/@#&]],{n,0,15}] (* Gus Wiseman, Oct 11 2023 *)

a(n) = A000041(2*n) - A000041(n).

Edited by Alois P. Heinz, Dec 03 2010

A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 1, 5, 3, 3, 8, 4, 9, 1, 17, 6, 16, 1, 2, 24, 7, 33, 4, 9, 46, 11, 52, 3, 18, 1, 4, 64, 12, 91, 6, 38, 3, 15, 1, 1, 107, 17, 138, 9, 68, 2, 28, 2, 12, 0, 2, 147, 19, 219, 12, 117, 6, 56, 3, 34, 2, 9, 0, 3

Offset: 0

Gus Wiseman, Nov 07 2023

			The partition (3,2,2,1) has two submultisets summing to 4, namely {2,2} and {1,3}, so it is counted under T(4,2).
The partition (2,2,1,1,1,1) has three submultisets summing to 4, namely {1,1,1,1}, {1,1,2}, and {2,2}, so it is counted under T(4,3).
Triangle begins:
    0   1
    1   1
    2   2   1
    5   3   3
    8   4   9   1
   17   6  16   1   2
   24   7  33   4   9
   46  11  52   3  18   1   4
   64  12  91   6  38   3  15   1   1
  107  17 138   9  68   2  28   2  12   0   2
  147  19 219  12 117   6  56   3  34   2   9   0   3
Row n = 4 counts the following partitions:
  (8)     (44)        (431)      (221111)
  (71)    (3311)      (422)
  (62)    (2222)      (4211)
  (611)   (11111111)  (41111)
  (53)                (3221)
  (521)               (32111)
  (5111)              (311111)
  (332)               (22211)
                      (2111111)

Row sums w/o the first column are A002219, ranks A357976, strict A237258.
Column k = 0 is A006827.
Row sums are A058696.
Column k = 1 is A108917.
The corresponding rank statistic is A357879 (without empty rows).
A000041 counts integer partitions, strict A000009.
A182616 counts partitions of 2n that do not contain n, ranks A366321.
A182616 counts partitions of 2n with at least one odd part, ranks A366530.
A276024 counts positive subset-sums of partitions, strict A284640.
A304792 counts subset-sums of partitions, rank statistic A299701.
A365543 counts partitions of n with a submultiset summing to k.
Cf. A046663, A064914, A079122, A122768, A213074, A231429, A235130, A237194.

t=Table[Length[Select[IntegerPartitions[2n], Count[Total/@Union[Subsets[#]],n]==k&]], {n,0,5}, {k,0,1+PartitionsP[n]}];
Table[NestWhile[Most,t[[i]],Last[#]==0&], {i,Length[t]}]

T(n,1) = A108917(n).

A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 33, 46, 68, 95, 140, 187, 266, 372, 507, 683, 948, 1256, 1692, 2263, 3003, 3955, 5248, 6824, 8921, 11669, 15058, 19413, 25128, 32149, 41129, 52578, 66740, 84696, 107389, 135310, 170277, 214386, 268151, 335261, 418896, 521204

Offset: 0

Wouter Meeussen, Jul 09 2005

nonn

Counted as orderless pairs since intersection is commutative.

			Of the partitions of 12 into different parts, the partition (5+4+2+1) has an empty intersection with only (12) and (9+3).
From _Gus Wiseman_, Oct 07 2023: (Start)
The a(6) = 4 pairs are:
  ((6),(5,1))
  ((6),(4,2))
  ((6),(3,2,1))
  ((5,1),(4,2))
(End)

Vaclav Kotesovec Table of n, a(n) for n = 0..1050 (terms 0..700 from Alois P. Heinz)

Column k=2 of A258280.
Cf. A079122, A086737.
Main diagonal of A284593 times (1/2).
This is the strict case of A260669.
The ordered version is A365662 = strict case of A054440.
This is the disjoint case of A366132, with twins A366317.
A000041 counts integer partitions, strict A000009.
A002219 counts biquanimous partitions, strict A237258, ordered A064914.
Cf. A000124, A000712, A001255, A006827, A032302, A122768, A355389.

using DiscreteMath`Combinatorica`and ListPartitionsQ[n_Integer]:= Flatten[ Reverse /@ Table[(Range[m-1, 0, -1]+#1&)/@ TransposePartition/@ Complement[Partitions[ n-m* (m-1)/2, m], Partitions[n-m*(m-1)/2, m-1]], {m, -1+Floor[1/2*(1+Sqrt[1+8*n])]}], 1]; Table[Plus@@Flatten[Outer[If[Intersection[Flatten[ #1], Flatten[ #2]]==={}, 1, 0]&, ListPartitionsQ[k], ListPartitionsQ[k], 1]], {k, 48}]/2
nmax = 50; p = 1; Do[p = Expand[p*(1 + x^j + y^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {j, 1, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Table[Coefficient[p, x^n*y^n]/2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 07 2017 *)
Table[Length[Select[Subsets[Select[IntegerPartitions[n], UnsameQ@@#&],{2}],Intersection@@#=={}&]],{n,15}] (* Gus Wiseman, Oct 07 2023 *)

a(n) = {my(A=1 + O(x*x^n) + O(y*y^n)); polcoef(polcoef(prod(k=1, n, A + x^k + y^k), n), n)/2} \\ Andrew Howroyd, Oct 10 2023

a(n) = ceiling(1/2 * [(x*y)^n] Product_{j>0} (1+x^j+y^j)). - Alois P. Heinz, Mar 31 2017

a(n) = ceiling(A365662(n)/2). - Gus Wiseman, Oct 07 2023

Name edited by Gus Wiseman, Oct 10 2023

a(0)=1 prepended by Alois P. Heinz, Feb 09 2024

A231429 Number of partitions of 2n into distinct parts < n.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 2, 4, 8, 14, 22, 35, 53, 78, 113, 160, 222, 306, 416, 558, 743, 980, 1281, 1665, 2149, 2755, 3514, 4458, 5626, 7070, 8846, 11020, 13680, 16920, 20852, 25618, 31375, 38309, 46649, 56651, 68616, 82908, 99940, 120192, 144238, 172730, 206425

Offset: 0

Reinhard Zumkeller, Nov 14 2013

nonn

From Gus Wiseman, Jun 17 2023: (Start)
Also the number of integer compositions of n with weighted sum 3*n, where the weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i * y_i. The a(0) = 1 through a(9) = 14 compositions are:
  ()  .  .  .  .  (11111)  (3111)   (3211)   (3311)    (3411)
                           (11211)  (11311)  (4121)    (4221)
                                    (12121)  (11411)   (5112)
                                    (21112)  (12221)   (11511)
                                             (13112)   (12321)
                                             (21131)   (13131)
                                             (21212)   (13212)
                                             (111122)  (21231)
                                                       (21312)
                                                       (22122)
                                                       (31113)
                                                       (111141)
                                                       (111222)
                                                       (112113)
For partitions we have A363527, ranks A363531. For reversed partitions we have A363526, ranks A363530.
(End)

			a(5) = #{4+3+2+1} = 1;
a(6) = #{5+4+3, 5+4+2+1} = 2;
a(7) = #{6+5+3, 6+5+2+1, 6+4+3+1, 5+4+3+2} = 4;
a(8) = #{7+6+3, 7+6+2+1, 7+6+3, 7+5+3+1, 7+4+3+2, 6+5+4+1, 6+5+3+2, 6+4+3+2+1} = 8;
a(9) = #{8+7+3, 8+7+2+1, 8+6+4, 8+6+3+1, 8+5+4+1, 8+5+3+2, 8+4+3+2+1, 7+6+5, 7+6+4+1, 7+6+3+2, 7+5+4+2, 7+5+3+2+1, 6+5+4+3, 6+5+4+2+1} = 14.

Cf. A209815, A079122.
A000041 counts integer partitions, strict A000009.
A053632 counts compositions by weighted sum.
A264034 counts partitions by weighted sum, reverse A358194.
A304818 gives weighted sum of prime indices, reverse A318283.
A320387 counts multisets by weighted sum, zero-based A359678.
Cf. A008284, A029931, A067538, A222855, A222955, A222970, A359042, A360672, A360675, A362559.

a231429 n = p [1..n-1] (2*n) where
   p _  0 = 1
   p [] _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], Total[Accumulate[#]]==3n&]],{n,0,15}] (* Gus Wiseman, Jun 17 2023 *)

A079124 Number of ways to partition n into distinct positive integers <= phi(n), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 1, 0, 1, 0, 2, 0, 4, 1, 5, 1, 11, 0, 17, 4, 13, 13, 37, 2, 53, 13, 51, 35, 103, 10, 135, 78, 167, 89, 255, 4, 339, 253, 378, 306, 542, 121, 759, 558, 872, 498, 1259, 121, 1609, 1180, 1677, 1665, 2589, 808, 3250, 1969, 3844, 3325, 5119, 1850, 6268, 4758, 7546, 7070

Offset: 0

Reinhard Zumkeller, Dec 27 2002

nonn

Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem, Mathematics and Computer Education, Vol. 31, No. 1, pp. 24-28, Winter 1997. MathEduc Database (Zentralblatt MATH, 1997c.01891).

Alois P. Heinz, Table of n, a(n) for n = 0..10000
Mohammad K. Azarian, A Generalization of the Climbing Stairs Problem II, Missouri Journal of Mathematical Sciences, Vol. 16, No. 1, Winter 2004, pp. 12-17. Zentralblatt MATH, Zbl 1071.05501.

Cf. A079126, A000009, A079122, A079125, A067953.

Cf. A227296, A036998.

a079124 n = p [1 .. a000010 n] n where
   p _      0 = 1
   p []     _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Jul 05 2013

with(numtheory):
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i-1))))
    end:
a:= n-> b(n, phi(n)):
seq(a(n), n=0..100);  # Alois P. Heinz, May 11 2015

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i-1]]]]; a[n_] := b[n, EulerPhi[n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

a(n) = b(0, n), b(m, n) = 1 + sum(b(i, j): m

a(0)=1 prepended by Alois P. Heinz, May 11 2015

A365828 Number of strict integer partitions of 2n not containing n.

Original entry on oeis.org

1, 1, 2, 3, 5, 8, 12, 18, 27, 39, 55, 78, 108, 148, 201, 270, 359, 475, 623, 811, 1050, 1351, 1728, 2201, 2789, 3517, 4418, 5527, 6887, 8553, 10585, 13055, 16055, 19685, 24065, 29343, 35685, 43287, 52387, 63253, 76200, 91605, 109897, 131575, 157231, 187539

Offset: 0

Gus Wiseman, Sep 20 2023

nonn

			The a(0) = 1 through a(6) = 12 strict partitions:
  ()  (2)  (4)    (6)    (8)      (10)       (12)
           (3,1)  (4,2)  (5,3)    (6,4)      (7,5)
                  (5,1)  (6,2)    (7,3)      (8,4)
                         (7,1)    (8,2)      (9,3)
                         (5,2,1)  (9,1)      (10,2)
                                  (6,3,1)    (11,1)
                                  (7,2,1)    (5,4,3)
                                  (4,3,2,1)  (7,3,2)
                                             (7,4,1)
                                             (8,3,1)
                                             (9,2,1)
                                             (5,4,2,1)

The complement is counted by A111133.
For non-strict partitions we have A182616, complement A000041.
A000009 counts strict integer partitions.
A046663 counts partitions with no submultiset summing to k, strict A365663.
A365827 counts strict partitions not of length 2, complement A140106.
Cf. A008967, A035363, A078408, A079122, A231429, A238628, A344415, A365659.

Table[Length[Select[IntegerPartitions[2n],UnsameQ@@#&&FreeQ[#,n]&]],{n,0,30}]

a(n) = A000009(2n) - A000009(n) + 1.

A079126 Triangle T(n,k) of numbers of partitions of n into distinct positive integers <= k, 0<=k<=n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 0, 0, 0, 1, 2, 3, 0, 0, 0, 1, 2, 3, 4, 0, 0, 0, 0, 2, 3, 4, 5, 0, 0, 0, 0, 1, 3, 4, 5, 6, 0, 0, 0, 0, 1, 3, 5, 6, 7, 8, 0, 0, 0, 0, 1, 3, 5, 7, 8, 9, 10, 0, 0, 0, 0, 0, 2, 5, 7, 9, 10, 11, 12, 0, 0, 0, 0, 0, 2, 5, 8, 10, 12, 13, 14, 15, 0, 0, 0, 0, 0, 1, 4, 8, 11, 13, 15, 16, 17, 18

Offset: 0

Reinhard Zumkeller, Dec 27 2002

T(n,n) = A000009(n), right side of the triangle;

T(n,k)=0 for n>0 and k < A002024(n); T(prime(n),n) = A067953(n) for n>0.

			The seven partitions of n=5 are {5}, {4,1}, {3,2}, {3,1,1}, {2,2,1}, {2,1,1,1} and {1,1,1,1,1}. Only two of them ({4,1} and {3,2}) have distinct parts <= 4, so T(5,4) = 2.
Triangle T(n,k) begins:
1;
0, 1;
0, 0, 1;
0, 0, 1, 2;
0, 0, 0, 1 ,2;
0, 0, 0, 1, 2, 3;
0, 0, 0, 1, 2, 3, 4;
0, 0, 0, 0, 2, 3, 4, 5;
0, 0, 0, 0, 1, 3, 4, 5,  6;
0, 0, 0, 0, 1, 3, 5, 6,  7,  8;
0, 0, 0, 0, 1, 3, 5, 7,  8,  9, 10;
0, 0, 0, 0, 0, 2, 5, 7,  9, 10, 11, 12;
0, 0, 0, 0, 0, 2, 5, 8, 10, 12, 13, 14, 15; ...

Alois P. Heinz, Rows n = 0..140, flattened
Eric Weisstein's World of Mathematics, Partition Function Q.

Cf. A000009, A079122, A035294, A079124, A079125.

Differs from A026840 in having extra zeros at the ends of the rows.

T:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, T(n, i-1)+`if`(i>n, 0, T(n-i, i-1))))
    end:
seq(seq(T(n,k), k=0..n), n=0..20);  # Alois P. Heinz, May 11 2015

T[n_, i_] := T[n, i] = If[n==0, 1, If[i<1, 0, T[n, i-1] + If[i>n, 0, T[n-i, i-1]]]]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 20}] // Flatten (* Jean-François Alcover, Jun 30 2015, after Alois P. Heinz *)

T(n,k) = b(0,n,k), where b(m,n,k) = 1+sum(b(i,j,k): m

T(n,k) = coefficient of x^n in product_{i=1..k} (1+x^i). - Vladeta Jovovic, Aug 07 2003

A079125 Number of ways to partition the sum of all divisors of n (sigma(n), A000203) into distinct positive integers not greater than n.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 5, 13, 13, 27, 14, 89, 21, 89, 97, 230, 45, 613, 63, 980, 347, 580, 121, 6663, 330, 1289, 1043, 5847, 295, 26488, 389, 12813, 2800, 5411, 2840, 156304, 863, 10433, 6939, 161711, 1425, 272499, 1815, 103738, 61469, 35448, 2909, 2475011

Offset: 1

Reinhard Zumkeller, Dec 27 2002

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A079126, A000009, A079122, A079124, A067953.

a[n_] := Module[{sum = DivisorSigma[1, n], x}, CoefficientList[Product[1 + x^i, {i, 1, n}], x][[1 + sum]]]; Array[a, 50] (* Amiram Eldar, Apr 15 2025 *)

a(n)=my(v=partitions(sigma(n),n));sum(i=1,#v,#vecsort(v[i],,8)==#v[i]) \\ Charles R Greathouse IV, Feb 14 2013

a(n) = b(0, n), b(m, n) = 1 + Sum_{m < i < j < n & i+j = sigma(n)} b(i, j).

Showing 1-10 of 10 results.

cp's OEIS Frontend

A209816 Number of partitions of 2n in which every part is

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A035294 Number of ways to partition 2n into distinct positive integers.

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A182616 Number of partitions of 2n that contain odd parts.

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A367094 Irregular triangle read by rows with trailing zeros removed where T(n,k) is the number of integer partitions of 2n whose number of submultisets summing to n is k.

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A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

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A231429 Number of partitions of 2n into distinct parts < n.

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A079124 Number of ways to partition n into distinct positive integers <= phi(n), where phi is Euler's totient function (A000010).

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