A305551 -id:A305551 - cp's OEIS frontend

A306017 Number of non-isomorphic multiset partitions of weight n in which all parts have the same size.

1, 1, 4, 6, 17, 14, 66, 30, 189, 222, 550, 112, 4696, 202, 5612, 30914, 63219, 594, 453125, 980, 3602695, 5914580, 1169348, 2510, 299083307, 232988061, 23248212, 2669116433, 14829762423, 9130, 170677509317, 13684, 1724710753084, 2199418340875, 14184712185, 38316098104262

Offset: 0

Gus Wiseman, Jun 17 2018

nonn

A multiset partition of weight n is a finite multiset of finite nonempty multisets whose sizes sum to n.
Number of distinct nonnegative integer matrices with all row sums equal and total sum n up to row and column permutations. - Andrew Howroyd, Sep 05 2018
From Gus Wiseman, Oct 11 2018: (Start)
Also the number of non-isomorphic multiset partitions of weight n in which each vertex appears the same number of times. For n = 4, non-isomorphic representatives of these 17 multiset partitions are:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,3,4}}
  {{1},{1,1,1}}
  {{1},{1,2,2}}
  {{1},{2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{3,4}}
  {{1},{1},{1,1}}
  {{1},{1},{2,2}}
  {{1},{2},{1,2}}
  {{1},{2},{3,4}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}
  {{1},{2},{3},{4}}
(End)

			Non-isomorphic representatives of the a(4) = 17 multiset partitions:
  {{1,1,1,1}}
  {{1,1,2,2}}
  {{1,2,2,2}}
  {{1,2,3,3}}
  {{1,2,3,4}}
  {{1,1},{1,1}}
  {{1,1},{2,2}}
  {{1,2},{1,2}}
  {{1,2},{2,2}}
  {{1,2},{3,3}}
  {{1,2},{3,4}}
  {{1,3},{2,3}}
  {{1},{1},{1},{1}}
  {{1},{1},{2},{2}}
  {{1},{2},{2},{2}}
  {{1},{2},{3},{3}}
  {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000005, A001315, A007716, A038041, A049311, A283877, A298422, A306018, A306019, A306020, A306021, A318951.

Cf. A064573, A279787, A305551, A319616, A319056, A331485.

permcount[v_List] := Module[{m = 1, s = 0, k = 0, t}, For[i = 1, i <= Length[v], i++, t = v[[i]]; k = If[i > 1 && t == v[[i - 1]], k + 1, 1]; m *= t*k; s += t]; s!/m];
K[q_List, t_, k_] := SeriesCoefficient[1/Product[g = GCD[t, q[[j]]]; (1 - x^(q[[j]]/g))^g, {j, 1, Length[q]}], {x, 0, k}];
RowSumMats[n_, m_, k_] := Module[{s = 0}, Do[s += permcount[q]* SeriesCoefficient[Exp[Sum[K[q, t, k]/t*x^t, {t, 1, n}]], {x, 0, n}], {q, IntegerPartitions[m]}]; s/m!];
a[n_] := a[n] = If[n==0, 1, If[PrimeQ[n], 2 PartitionsP[n], Sum[ RowSumMats[ n/d, n, d], {d, Divisors[n]}]]];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 35}] (* Jean-François Alcover, Nov 07 2019, after Andrew Howroyd *)

\\ See A318951 for RowSumMats.
a(n)={sumdiv(n,d,RowSumMats(n/d,n,d))} \\ Andrew Howroyd, Sep 05 2018

For p prime, a(p) = 2*A000041(p).

a(n) = Sum_{d|n} A331485(n/d, d). - Andrew Howroyd, Feb 09 2020

Terms a(11) and beyond from Andrew Howroyd, Sep 05 2018

A006827 Number of partitions of 2n with all subsums different from n.

Original entry on oeis.org

1, 2, 5, 8, 17, 24, 46, 64, 107, 147, 242, 302, 488, 629, 922, 1172, 1745, 2108, 3104, 3737, 5232, 6419, 8988, 10390, 14552, 17292, 23160, 27206, 36975, 41945, 57058, 65291, 85895, 99384, 130443, 145283, 193554, 218947, 281860, 316326, 413322, 454229, 594048

Offset: 1

N. J. A. Sloane

Partitions of this type are also called non-biquanimous partitions. - Gus Wiseman, Apr 19 2024

			From _Gus Wiseman_, Apr 19 2024: (Start)
The a(1) = 1 through a(5) = 17 partitions (A = 10):
  (2)  (4)   (6)    (8)     (A)
       (31)  (42)   (53)    (64)
             (51)   (62)    (73)
             (222)  (71)    (82)
             (411)  (332)   (91)
                    (521)   (433)
                    (611)   (442)
                    (5111)  (622)
                            (631)
                            (721)
                            (811)
                            (3331)
                            (4222)
                            (6211)
                            (7111)
                            (22222)
                            (61111)
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Fausto A. C. Cariboni, Table of n, a(n) for n = 1..140 (terms 1..89 from Alois P. Heinz)
P. Erdős, J. L. Nicolas and A. Sárközy, On the number of partitions of n without a given subsum (I), Discrete Math., 75 (1989), 155-166 = Annals Discrete Math. Vol. 43, Graph Theory and Combinatorics 1988, ed. B. Bollobas.

The complement is counted by A002219, ranks A357976.
Central diagonal of A046663.
The strict case is A321142, even bisection of A371794 (odd A078408).
This is the "bi-" version of A321451, ranks A321453.
Column k = 0 of A367094.
These partitions have Heinz numbers A371731.
Even bisection of A371795 (odd A058695).
A371783 counts k-quanimous partitions.
Cf. A035470, A064914, A237258, A305551, A321452, A365543, A365663, A366320, A371736, A371782, A371792.

b:= proc(n, i, s) option remember;
      `if`(0 in s or n in s, 0, `if`(n=0, 1, `if`(i<1, 0, b(n, i-1, s)+
      `if`(i<=n, b(n-i, i, select(y-> 0<=y and y<=n-i,
                 map(x-> [x, x-i][], s))), 0))))
    end:
a:= n-> b(2*n, 2*n, {n}):
seq(a(n), n=1..25);  # Alois P. Heinz, Jul 10 2012

b[n_, i_, s_] := b[n, i, s] = If[MemberQ[s, 0 | n], 0, If[n == 0, 1, If[i < 1, 0, b[n, i-1, s] + If[i <= n, b[n-i, i, Select[Flatten[Transpose[{s, s-i}]], 0 <= # <= n-i &]], 0]]]]; a[n_] := b[2*n, 2*n, {n}]; Table[Print[an = a[n]]; an, {n, 1, 25}] (* Jean-François Alcover, Nov 12 2013, after Alois P. Heinz *)

from itertools import combinations_with_replacement
from collections import Counter
from sympy import npartitions
from sympy.utilities.iterables import partitions
def A006827(n): return npartitions(n<<1)-len({tuple(sorted((p+q).items())) for p, q in combinations_with_replacement(tuple(Counter(p) for p in partitions(n)),2)}) # Chai Wah Wu, Sep 20 2023

a(n) = A000041(2*n) - A002219(n).

a(n) = A046663(2*n,n).

More terms from Don Reble, Nov 03 2001

More terms from Alois P. Heinz, Jul 10 2012

A321455 Number of ways to factor n into factors > 1 all having the same sum of prime indices.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3

Offset: 1

Gus Wiseman, Nov 10 2018

nonn

Also the number of multiset partitions of the multiset of prime indices of n with equal block-sums.

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum of prime indices of n is A056239(n).

			The a(1440) = 6 factorizations into factors all having the same sum of prime indices:
  (10*12*12)
  (5*6*6*8)
  (9*10*16)
  (30*48)
  (36*40)
  (1440)
The a(900) = 5 multiset partitions with equal block-sums:
  {{1,1,2,2,3,3}}
  {{3,3},{1,1,2,2}}
  {{1,2,3},{1,2,3}}
  {{1,3},{1,3},{2,2}}
  {{3},{3},{1,2},{1,2}}

Positions of 1's are A321453. Positions of terms > 1 are A321454.

Cf. A001055, A035470, A056239, A279787, A305551, A321469, A322794, A326515, A326516, A326518, A326534.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],SameQ@@hwt/@#&]],{n,100}]

A056239(n) = if(1==n, 0, my(f=factor(n)); sum(i=1, #f~, f[i, 2] * primepi(f[i, 1])));
all_have_same_sum_of_pis(facs) = if(!#facs, 1, (#Set(apply(A056239,facs)) == 1));
A321455(n, m=n, facs=List([])) = if(1==n, all_have_same_sum_of_pis(facs), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A321455(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Jan 20 2025

Data section extended to a(108) by Antti Karttunen, Jan 20 2025

A261049 Expansion of Product_{k>=1} (1+x^k)^(p(k)), where p(k) is the partition function.

Original entry on oeis.org

1, 1, 2, 5, 9, 19, 37, 71, 133, 252, 464, 851, 1547, 2787, 4985, 8862, 15639, 27446, 47909, 83168, 143691, 247109, 423082, 721360, 1225119, 2072762, 3494359, 5870717, 9830702, 16409939, 27309660, 45316753, 74986921, 123748430, 203686778, 334421510, 547735241

Offset: 0

Vaclav Kotesovec, Aug 08 2015

nonn

Number of strict multiset partitions of integer partitions of n. Weigh transform of A000041. - Gus Wiseman, Oct 11 2018

			From _Gus Wiseman_, Oct 11 2018: (Start)
The a(1) = 1 through a(5) = 19 strict multiset partitions:
  {{1}}  {{2}}    {{3}}        {{4}}          {{5}}
         {{1,1}}  {{1,2}}      {{1,3}}        {{1,4}}
                  {{1,1,1}}    {{2,2}}        {{2,3}}
                  {{1},{2}}    {{1,1,2}}      {{1,1,3}}
                  {{1},{1,1}}  {{1},{3}}      {{1,2,2}}
                               {{1,1,1,1}}    {{1},{4}}
                               {{1},{1,2}}    {{2},{3}}
                               {{2},{1,1}}    {{1,1,1,2}}
                               {{1},{1,1,1}}  {{1},{1,3}}
                                              {{1},{2,2}}
                                              {{2},{1,2}}
                                              {{3},{1,1}}
                                              {{1,1,1,1,1}}
                                              {{1},{1,1,2}}
                                              {{1,1},{1,2}}
                                              {{2},{1,1,1}}
                                              {{1},{1,1,1,1}}
                                              {{1,1},{1,1,1}}
                                              {{1},{2},{1,1}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..1000
R. Kaneiwa, An asymptotic formula for Cayley's double partition function p(2; n), Tokyo J. Math. 2, 137-158 (1979).

Cf. A000041, A001970, A026007, A027998, A248882, A102866, A256142.
Cf. A047968, A050342, A089259, A305551, A320328, A320330, A320331.
Row sums of A360742.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      binomial(combinat[numbpart](i), j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 08 2015

nmax=40; CoefficientList[Series[Product[(1+x^k)^PartitionsP[k],{k,1,nmax}],{x,0,nmax}],x]

A319066 Number of partitions of integer partitions of n where all parts have the same length.

Original entry on oeis.org

1, 1, 3, 5, 10, 14, 26, 35, 59, 82, 128, 176, 273, 371, 553, 768, 1119, 1544, 2235, 3084, 4410, 6111, 8649, 11982, 16901, 23383, 32780, 45396, 63365, 87622, 121946, 168407, 233605, 322269, 445723, 613922, 847131, 1164819, 1603431, 2201370, 3023660, 4144124, 5680816

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

			The a(1) = 1 through a(5) = 14 multiset partitions:
  {{1}}  {{2}}      {{3}}          {{4}}              {{5}}
         {{1,1}}    {{1,2}}        {{1,3}}            {{1,4}}
         {{1},{1}}  {{1,1,1}}      {{2,2}}            {{2,3}}
                    {{1},{2}}      {{1,1,2}}          {{1,1,3}}
                    {{1},{1},{1}}  {{1},{3}}          {{1,2,2}}
                                   {{2},{2}}          {{1},{4}}
                                   {{1,1,1,1}}        {{2},{3}}
                                   {{1,1},{1,1}}      {{1,1,1,2}}
                                   {{1},{1},{2}}      {{1,1,1,1,1}}
                                   {{1},{1},{1},{1}}  {{1,1},{1,2}}
                                                      {{1},{1},{3}}
                                                      {{1},{2},{2}}
                                                      {{1},{1},{1},{2}}
                                                      {{1},{1},{1},{1},{1}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A001970, A047968, A261049, A279787, A305551, A306017, A319056.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[Join@@mps/@IntegerPartitions[n],SameQ@@Length/@#&]],{n,8}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(p=1/prod(k=1, n, 1 - x^k*y + O(x*x^n))); concat([1], sum(k=1, n, EulerT(Vec(polcoef(p, k, y), -n))))} \\ Andrew Howroyd, Oct 25 2018

Terms a(11) and beyond from Andrew Howroyd, Oct 25 2018

A320322 Number of integer partitions of n whose product is a perfect power.

Original entry on oeis.org

1, 0, 0, 0, 2, 2, 5, 5, 9, 11, 18, 19, 28, 30, 42, 50, 68, 76, 102, 113, 146, 170, 212, 241, 312, 356, 441, 514, 628, 720, 887, 1008, 1215, 1403, 1660, 1903, 2291, 2609, 3107, 3594, 4254, 4864, 5739, 6546, 7672, 8811, 10237, 11651, 13583, 15420, 17867, 20382

Offset: 0

Gus Wiseman, Oct 10 2018

nonn

			The a(4) = 2 through a(11) = 19 integer partitions:
  4   41   33    331    8       9        55        551
  22  221  42    421    44      81       82        632
           222   2221   422     333      91        821
           411   4111   2222    441      433       911
           2211  22111  3311    4221     442       4331
                        4211    22221    811       4421
                        22211   33111    3322      8111
                        41111   42111    3331      33221
                        221111  222111   4222      33311
                                411111   4411      42221
                                2211111  22222     44111
                                         42211     222221
                                         222211    422111
                                         331111    2222111
                                         421111    3311111
                                         2221111   4211111
                                         4111111   22211111
                                         22111111  41111111
                                                   221111111

Cf. A003963, A064573, A305551, A319056, A319066, A319071, A319169, A320325.

Table[Length[Select[IntegerPartitions[n],GCD@@FactorInteger[Times@@#][[All,2]]>1&]],{n,30}]

A321452 Number of integer partitions of n that can be partitioned into two or more blocks with equal sums.

Original entry on oeis.org

0, 0, 1, 1, 3, 1, 7, 1, 14, 10, 26, 1, 55, 1, 90, 68, 167, 1, 292, 1, 482, 345, 761, 1, 1291, 266, 1949, 1518, 3091, 1, 4793, 1, 7177, 5612, 10566, 2623, 16007, 1, 22912, 18992, 33619, 1, 48529, 1, 68758, 59187, 96571, 1, 137489, 11418, 189979, 167502, 264299

Offset: 0

Gus Wiseman, Nov 10 2018

nonn

a(n) = 1 if and only if n is prime. - Chai Wah Wu, Nov 12 2018

			The a(2) = 1 through a(9) = 10 partitions:
  (11)  (111)  (22)    (11111)  (33)      (1111111)  (44)        (333)
               (211)            (222)                (422)       (3321)
               (1111)           (321)                (431)       (32211)
                                (2211)               (2222)      (33111)
                                (3111)               (3221)      (222111)
                                (21111)              (3311)      (321111)
                                (111111)             (4211)      (2211111)
                                                     (22211)     (3111111)
                                                     (32111)     (21111111)
                                                     (41111)     (111111111)
                                                     (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)
The partition (32111) can be partitioned as ((13)(112)), and the blocks both sum to 4, so (32111) is counted under a(8).

Cf. A000041, A265947, A276024, A279787, A305551, A306017, A317141, A320322, A321451, A321453, A321454, A321455.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[facs[Times@@Prime/@#],SameQ@@hwt/@#&]]>1&]],{n,10}]

a(n) = A000041(n) - A321451(n).

a(26)-a(52) from Alois P. Heinz, Nov 11 2018

A321451 Number of integer partitions of n that cannot be partitioned into two or more blocks with equal sums.

Original entry on oeis.org

1, 1, 1, 2, 2, 6, 4, 14, 8, 20, 16, 55, 22, 100, 45, 108, 64, 296, 93, 489, 145, 447, 241, 1254, 284, 1692, 487, 1492, 627, 4564, 811, 6841, 1172, 4531, 1744, 12260, 1970, 21636, 3103, 12193, 3719, 44582, 4645, 63260, 6417, 29947, 8987, 124753, 9784, 162107, 14247

Offset: 0

Gus Wiseman, Nov 10 2018

nonn

			The a(1) = 1 through a(9) = 20 partitions:
  (1)  (2)  (3)   (4)   (5)     (6)    (7)       (8)     (9)
            (21)  (31)  (32)    (42)   (43)      (53)    (54)
                        (41)    (51)   (52)      (62)    (63)
                        (221)   (411)  (61)      (71)    (72)
                        (311)          (322)     (332)   (81)
                        (2111)         (331)     (521)   (432)
                                       (421)     (611)   (441)
                                       (511)     (5111)  (522)
                                       (2221)            (531)
                                       (3211)            (621)
                                       (4111)            (711)
                                       (22111)           (3222)
                                       (31111)           (4221)
                                       (211111)          (4311)
                                                         (5211)
                                                         (6111)
                                                         (22221)
                                                         (42111)
                                                         (51111)
                                                         (411111)
A complete list of all multiset partitions of the partition (2111) into two or more blocks is: ((1)(112)), ((2)(111)), ((11)(12)), ((1)(1)(12)), ((1)(2)(11)), ((1)(1)(1)(2)). None of these has equal block-sums, so (2111) is counted toward a(5).
On the other hand, the partition (321) can be partitioned as ((12)(3)), which has two or more blocks and equal block-sums, so (321) is not counted toward a(6).

Cf. A000041, A265947, A276024, A279787, A305551, A306017, A317141, A320322, A321452, A321453, A321454, A321455.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[IntegerPartitions[n],Length[Select[facs[Times@@Prime/@#],SameQ@@hwt/@#&]]==1&]],{n,10}]

a(n) = A000041(n) - A321452(n).

a(33)-a(50) from Alois P. Heinz, Nov 11 2018

A321454 Numbers that can be factored into two or more factors all having the same sum of prime indices.

Original entry on oeis.org

4, 8, 9, 12, 16, 25, 27, 30, 32, 36, 40, 48, 49, 63, 64, 70, 81, 84, 90, 100, 108, 112, 120, 121, 125, 128, 144, 150, 154, 160, 165, 169, 180, 192, 196, 198, 200, 210, 216, 220, 225, 240, 243, 252, 256, 264, 270, 273, 280, 286, 288, 289, 300, 320, 324, 325

Offset: 1

Gus Wiseman, Nov 10 2018

nonn

Also Heinz numbers of integer partitions that can be partitioned into two or more blocks with equal sums. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The sum of prime indices of n is A056239(n).

			The sequence of all integer partitions that can be partitioned into two or more blocks with equal sums begins: (11), (111), (22), (211), (1111), (33), (222), (321), (11111), (2211), (3111), (21111), (44), (422), (111111), (431), (2222), (4211), (3221), (3311), (22211), (41111), (32111), (55), (333), (1111111), (221111), (3321), (541), (311111), (532), (66), (32211), (2111111), (4411), (5221), (33111).
The Heinz number of (32111) is 120, which has factorization (10*12) corresponding to the multiset partition ((13)(112)) whose blocks have equal sums, so 120 belongs to the sequence.

Positions of terms > 1 in A321455.

Cf. A056239, A112798, A276024, A279787, A305551, A306017, A317144, A320322, A321451, A321452, A321453.

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[facs[#],And[Length[#]>1,SameQ@@hwt/@#]&]!={}&]

A321142 Number of strict integer partitions of 2*n with no subset summing to n.

Original entry on oeis.org

0, 1, 2, 3, 5, 7, 11, 15, 23, 30, 43, 57, 79, 102, 138, 174, 232, 292, 375, 471, 602, 741, 935, 1148, 1425, 1733, 2137, 2571, 3156, 3789, 4557, 5470, 6582, 7796, 9317, 11027, 13058, 15400, 18159, 21249, 24971, 29170, 33986, 39596, 46073, 53219, 61711, 71330, 82171

Offset: 0

Gus Wiseman, Oct 28 2018

nonn

			The a(1) = 1 through a(8) = 23 partitions:
  (2)  (4)    (6)    (8)      (10)     (12)     (14)       (16)
       (3,1)  (4,2)  (5,3)    (6,4)    (7,5)    (8,6)      (9,7)
              (5,1)  (6,2)    (7,3)    (8,4)    (9,5)      (10,6)
                     (7,1)    (8,2)    (9,3)    (10,4)     (11,5)
                     (5,2,1)  (9,1)    (10,2)   (11,3)     (12,4)
                              (6,3,1)  (11,1)   (12,2)     (13,3)
                              (7,2,1)  (5,4,3)  (13,1)     (14,2)
                                       (7,3,2)  (6,5,3)    (15,1)
                                       (7,4,1)  (8,4,2)    (7,5,4)
                                       (8,3,1)  (8,5,1)    (7,6,3)
                                       (9,2,1)  (9,3,2)    (9,4,3)
                                                (9,4,1)    (9,5,2)
                                                (10,3,1)   (9,6,1)
                                                (11,2,1)   (10,4,2)
                                                (8,3,2,1)  (10,5,1)
                                                           (11,3,2)
                                                           (11,4,1)
                                                           (12,3,1)
                                                           (13,2,1)
                                                           (6,5,4,1)
                                                           (7,4,3,2)
                                                           (9,4,2,1)
                                                           (10,3,2,1)

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..200

Cf. A002219, A006827, A108917, A237258, A275972, A279791, A284640, A305551.

Table[Length[Select[IntegerPartitions[n],And[UnsameQ@@#,!Or@@Table[SameQ[Total[#[[s]]],n/2],{s,Subsets[Range[Length[#]]]}]]&]],{n,2,20,2}]

a(33)-a(48) from Giovanni Resta, Oct 30 2018

cp's OEIS Frontend

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