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%I #13 Jan 15 2024 20:29:22
%S 1,1,0,4,7,16,52,206,444,1624,5462,19188,62890,215367,765694,2854202,
%T 10634247,39842786,150669765,581189458,2287298588,9157598354,
%U 37109364812,151970862472,629048449881,2635589433705,11184718653563,48064965080106,208988724514022,918639253237646,4079974951494828
%N Number of non-isomorphic balanced multiset partitions of weight n.
%C We define a multiset partition to be balanced if it has exactly as many parts as the greatest size of a part.
%e Non-isomorphic representatives of the a(1) = 1 through a(5) = 16 multiset partitions (empty column indicated by dot):
%e {{1}} . {{1},{1,1}} {{1,1},{1,1}} {{1},{1},{1,1,1}}
%e {{1},{2,2}} {{1,1},{2,2}} {{1},{1},{1,2,2}}
%e {{1},{2,3}} {{1,2},{1,2}} {{1},{1},{2,2,2}}
%e {{2},{1,2}} {{1,2},{2,2}} {{1},{1},{2,3,3}}
%e {{1,2},{3,3}} {{1},{1},{2,3,4}}
%e {{1,2},{3,4}} {{1},{2},{1,2,2}}
%e {{1,3},{2,3}} {{1},{2},{2,2,2}}
%e {{1},{2},{2,3,3}}
%e {{1},{2},{3,3,3}}
%e {{1},{2},{3,4,4}}
%e {{1},{2},{3,4,5}}
%e {{1},{3},{2,3,3}}
%e {{1},{4},{2,3,4}}
%e {{2},{2},{1,2,2}}
%e {{2},{3},{1,2,3}}
%e {{3},{3},{1,2,3}}
%o (PARI) \\ See A340652 for G.
%o seq(n)={Vec(1 + sum(k=1,n,polcoef(G(n,n,k,y),k,y) - polcoef(G(n,n,k-1,y),k,y)))} \\ _Andrew Howroyd_, Jan 15 2024
%Y The version for partitions is A047993.
%Y The co-balanced version is A319616.
%Y The cross-balanced version is A340651.
%Y The twice-balanced version is A340652.
%Y The version for factorizations is A340653.
%Y A007716 counts non-isomorphic multiset partitions.
%Y A007718 counts non-isomorphic connected multiset partitions.
%Y A316980 counts non-isomorphic strict multiset partitions.
%Y Other balance-related sequences:
%Y - A098124 counts balanced compositions.
%Y - A106529 lists balanced numbers.
%Y - A340596 counts co-balanced factorizations.
%Y - A340597 lists numbers with an alt-balanced factorization.
%Y - A340598 counts balanced set partitions.
%Y - A340599 counts alt-balanced factorizations.
%Y Cf. A001055, A006141, A007717, A316983, A320663, A324522, A340654, A340655.
%K nonn
%O 0,4
%A _Gus Wiseman_, Feb 05 2021
%E a(11) onwards from _Andrew Howroyd_, Jan 15 2024
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