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%I #16 Jan 11 2020 20:44:27
%S 1,2,2,5,3,15,9,5,52,7,203,66,31,11,877,15,4140,712,109,22,21147,686,
%T 30,115975,10457,339,42,678570,56,4213597,198091,27036,6721,1043,77,
%U 27644437,101,190899322,4659138,2998,135,1382958545,1688360,58616,176
%N Irregular triangle read by rows where if d|n then T(n,d) is the number of multiset partitions of a multiset with d copies of each integer from 1 to n/d.
%H Andrew Howroyd, <a href="/A322786/b322786.txt">Table of n, a(n) for n = 1..207</a> (first 50 rows)
%F T(n,d) = A001055(A002110(n/d)^d).
%F T(n,d) = A219727(d, n/d). - _Andrew Howroyd_, Jan 11 2020
%e Triangle begins:
%e 1
%e 2 2
%e 5 3
%e 15 9 5
%e 52 7
%e 203 66 31 11
%e 877 15
%e 4140 712 109 22
%e 21147 686 30
%e 115975 10457 339 42
%e 678570 56
%e 4213597 198091 27036 6721 1043 77
%e For example, row 4 counts the following multiset partitions.
%e {{1,2,3,4}} {{1,1,2,2}} {{1,1,1,1}}
%e {{1},{2,3,4}} {{1},{1,2,2}} {{1},{1,1,1}}
%e {{1,2},{3,4}} {{1,1},{2,2}} {{1,1},{1,1}}
%e {{1,3},{2,4}} {{1,2},{1,2}} {{1},{1},{1,1}}
%e {{1,4},{2,3}} {{2},{1,1,2}} {{1},{1},{1},{1}}
%e {{2},{1,3,4}} {{1},{1},{2,2}}
%e {{3},{1,2,4}} {{1},{2},{1,2}}
%e {{4},{1,2,3}} {{2},{2},{1,1}}
%e {{1},{2},{3,4}} {{1},{1},{2},{2}}
%e {{1},{3},{2,4}}
%e {{1},{4},{2,3}}
%e {{2},{3},{1,4}}
%e {{2},{4},{1,3}}
%e {{3},{4},{1,2}}
%e {{1},{2},{3},{4}}
%t u[n_,k_]:=u[n,k]=If[n==1,1,Sum[u[n/d,d],{d,Select[Rest[Divisors[n]],#<=k&]}]];
%t Table[Table[u[Array[Prime,n/d,1,Times]^d,Array[Prime,n/d,1,Times]^d],{d,Divisors[n]}],{n,10}]
%o (PARI) \\ needs T(n,k) from A219727.
%o Row(n)={[T(d,n/d) | d<-divisors(n)]}
%o { for(n=1, 12, print(Row(n))) } \\ _Andrew Howroyd_, Jan 11 2020
%Y Row sums are A322784. First column is A000110.
%Y Cf. A001055, A005176, A056239, A072774, A100778, A219727, A295193, A306017, A319190, A319612, A322784, A322785, A322787, A322788, A322792.
%K nonn,tabf
%O 1,2
%A _Gus Wiseman_, Dec 26 2018
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