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%I #8 Jan 15 2024 16:29:44
%S 1,0,1,0,2,1,0,3,2,1,0,5,8,3,1,0,7,17,12,3,1,0,11,46,45,18,4,1,0,15,
%T 94,141,76,23,4,1,0,22,212,432,333,124,30,5,1,0,30,416,1231,1254,622,
%U 178,37,5,1,0,42,848,3346,4601,2914,1058,252,45,6,1
%N Regular triangle read by rows where T(n,k) is the number of non-isomorphic connected multiset partitions of weight n with k vertices.
%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="/A322133/b322133.txt">Table of n, a(n) for n = 0..1325</a> (rows 0..50)
%e Triangle begins:
%e 1
%e 0 1
%e 0 2 1
%e 0 3 2 1
%e 0 5 8 3 1
%e 0 7 17 12 3 1
%e 0 11 46 45 18 4 1
%e 0 15 94 141 76 23 4 1
%e 0 22 212 432 333 124 30 5 1
%e 0 30 416 1231 1254 622 178 37 5 1
%e 0 42 848 3346 4601 2914 1058 252 45 6 1
%e Non-isomorphic representatives of the multiset partitions counted in row 4:
%e {{1,1,1,1}} {{1,1,2,2}} {{1,2,3,3}} {{1,2,3,4}}
%e {{1},{1,1,1}} {{1,2,2,2}} {{1,3},{2,3}}
%e {{1,1},{1,1}} {{1},{1,2,2}} {{3},{1,2,3}}
%e {{1},{1},{1,1}} {{1,2},{1,2}}
%e {{1},{1},{1},{1}} {{1,2},{2,2}}
%e {{2},{1,2,2}}
%e {{1},{2},{1,2}}
%e {{2},{2},{1,2}}
%o (PARI) \\ Needs G(m,n) defined in A317533 (faster PARI).
%o InvEulerMTS(p)={my(n=serprec(p, x)-1, q=log(p), vars=variables(p)); sum(i=1, n, moebius(i)*substvec(q + O(x*x^(n\i)), vars, apply(v->v^i, vars))/i)}
%o T(n)={[Vecrev(p) | p <- Vec(1 + InvEulerMTS(y^n*G(n,n) + sum(k=0, n-1, y^k*(1 - y)*G(k,n))))]}
%o { my(A=T(10)); for(i=1, #A, print(A[i])) } \\ _Andrew Howroyd_, Jan 15 2024
%Y Row sums are A007718.
%Y Cf. A000664, A007716, A054923, A191646, A191970, A275421, A286520, A317672, A319719, A321155, A321254, A322114.
%K nonn,tabl
%O 0,5
%A _Gus Wiseman_, Nov 27 2018
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