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%I #8 Jan 17 2023 18:20:39
%S 0,2,3,12,7,84,15,410,354,3073,56,28300,101,210036,126839,2070047,297,
%T 25295952,490,269662769,89071291,3449056162,1255,51132696310,
%U 400625539,713071048480,145126661415,11351097702297,4565,199926713003444,6842,3460838122540969
%N Number of non-isomorphic multiset partitions of weight n whose part-sizes have a common divisor > 1.
%C Also the number of nonnegative integer matrices up to row and column permutations with sum of elements equal to n and no zero rows or columns, in which the column sums are not relatively prime.
%C Also the number of non-isomorphic multiset partitions of weight n in which the multiset union of the parts is periodic, where a multiset is periodic if its multiplicities have a common divisor > 1.
%C The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.
%H Andrew Howroyd, <a href="/A320810/b320810.txt">Table of n, a(n) for n = 1..50</a>
%F a(n) = A007716(n) - A321283(n). - _Andrew Howroyd_, Jan 17 2023
%e Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions whose part-sizes have a common divisor:
%e {{1,1}} {{1,1,1}} {{1,1,1,1}} {{1,1,1,1,1}}
%e {{1,2}} {{1,2,2}} {{1,1,2,2}} {{1,1,2,2,2}}
%e {{1,2,3}} {{1,2,2,2}} {{1,2,2,2,2}}
%e {{1,2,3,3}} {{1,2,2,3,3}}
%e {{1,2,3,4}} {{1,2,3,3,3}}
%e {{1,1},{1,1}} {{1,2,3,4,4}}
%e {{1,1},{2,2}} {{1,2,3,4,5}}
%e {{1,2},{1,2}}
%e {{1,2},{2,2}}
%e {{1,2},{3,3}}
%e {{1,2},{3,4}}
%e {{1,3},{2,3}}
%e Non-isomorphic representatives of the a(2) = 1 through a(5) = 7 multiset partitions with periodic multiset union:
%e {{1,1}} {{1,1,1}} {{1,1,1,1}} {{1,1,1,1,1}}
%e {{1},{1}} {{1},{1,1}} {{1,1,2,2}} {{1},{1,1,1,1}}
%e {{1},{1},{1}} {{1},{1,1,1}} {{1,1},{1,1,1}}
%e {{1,1},{1,1}} {{1},{1},{1,1,1}}
%e {{1},{1,2,2}} {{1},{1,1},{1,1}}
%e {{1,1},{2,2}} {{1},{1},{1},{1,1}}
%e {{1,2},{1,2}} {{1},{1},{1},{1},{1}}
%e {{1},{1},{1,1}}
%e {{1},{1},{2,2}}
%e {{1},{2},{1,2}}
%e {{1},{1},{1},{1}}
%e {{1},{1},{2},{2}}
%o (PARI) \\ See links in A339645 for combinatorial species functions.
%o seq(n)={my(A=symGroupSeries(n));Vec(OgfSeries(sCartProd(sExp(A), -sum(d=2, n, moebius(d) * (-1 + sExp(O(x*x^n) + sum(i=1, n\d, polcoef(A,i*d)*x^(i*d)))) ))), -n)} \\ _Andrew Howroyd_, Jan 17 2023
%Y Cf. A007716, A018783, A047966, A120733, A303546, A303547, A305563, A319149, A319162, A319164, A319810.
%K nonn
%O 1,2
%A _Gus Wiseman_, Nov 15 2018
%E Terms a(11) and beyond from _Andrew Howroyd_, Jan 17 2023
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