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A358835 Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums.

%I A358835 #14 Mar 24 2025 05:29:24
%S A358835 1,1,3,4,8,8,17,16,31,34,54,57,108,102,166,191,294,298,504,491,803,
%T A358835 843,1251,1256,2167,1974,3133,3226,4972,4566,8018,6843,11657,11044,
%U A358835 17217,15010,28422,21638,38397,35067,58508,44584,91870,63262,125114,106264,177483
%N A358835 Number of multiset partitions of integer partitions of n with constant block sizes and constant block sums.
%H A358835 Andrew Howroyd, <a href="/A358835/b358835.txt">Table of n, a(n) for n = 0..1000</a>
%F A358835 a(n) = Sum_{d|n} Sum_{j=1..n/d} binomial(d + A008284(n/d, j) - 1, d) for n > 0. - _Andrew Howroyd_, Dec 31 2022
%e A358835 The a(1) = 1 through a(6) = 17 multiset partitions:
%e A358835   {1}  {2}     {3}        {4}           {5}              {6}
%e A358835        {11}    {12}       {13}          {14}             {15}
%e A358835        {1}{1}  {111}      {22}          {23}             {24}
%e A358835                {1}{1}{1}  {112}         {113}            {33}
%e A358835                           {1111}        {122}            {114}
%e A358835                           {2}{2}        {1112}           {123}
%e A358835                           {11}{11}      {11111}          {222}
%e A358835                           {1}{1}{1}{1}  {1}{1}{1}{1}{1}  {1113}
%e A358835                                                          {1122}
%e A358835                                                          {3}{3}
%e A358835                                                          {11112}
%e A358835                                                          {111111}
%e A358835                                                          {12}{12}
%e A358835                                                          {2}{2}{2}
%e A358835                                                          {111}{111}
%e A358835                                                          {11}{11}{11}
%e A358835                                                          {1}{1}{1}{1}{1}{1}
%t A358835 Table[If[n==0,1,Length[Union[Sort/@Join@@Table[Select[Tuples[IntegerPartitions[d],n/d],SameQ@@Length/@#&],{d,Divisors[n]}]]]],{n,0,20}]
%o A358835 (PARI)
%o A358835 P(n,y) = 1/prod(k=1, n, 1 - y*x^k + O(x*x^n))
%o A358835 seq(n) = {my(u=Vec(P(n,y)-1)); concat([1], vector(n, n, sumdiv(n, d, my(p=u[n/d]); sum(j=1, n/d, binomial(d + polcoef(p, j, y) - 1, d)))))} \\ _Andrew Howroyd_, Dec 31 2022
%Y A358835 For just constant sums we have A305551, ranked by A326534.
%Y A358835 For just constant lengths we have A319066, ranked by A320324.
%Y A358835 The version for set partitions is A327899.
%Y A358835 For distinct instead of constant lengths and sums we have A358832.
%Y A358835 The version for twice-partitions is A358833.
%Y A358835 A001970 counts multiset partitions of integer partitions.
%Y A358835 A063834 counts twice-partitions, strict A296122.
%Y A358835 Cf. A000219, A007425, A141199, A327908, A356932, A358831.
%K A358835 nonn
%O A358835 0,3
%A A358835 _Gus Wiseman_, Dec 05 2022
%E A358835 Terms a(41) and beyond from _Andrew Howroyd_, Dec 31 2022