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%I #17 Oct 22 2018 17:41:58
%S 0,0,1,0,0,1,0,0,0,2,0,0,0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,1,0,0,0,0,0,2,
%T 0,1,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,1,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,
%U 0,0,0,0,2,1,0,0,0,1,0,0,0,0,0,0,0,0,0
%N Regular triangle where T(n,k) is the number of factorizations of n into factors > 1 with sum k.
%e Triangle begins:
%e 0
%e 0 1
%e 0 0 1
%e 0 0 0 2
%e 0 0 0 0 1
%e 0 0 0 0 1 1
%e 0 0 0 0 0 0 1
%e 0 0 0 0 0 2 0 1
%e 0 0 0 0 0 1 0 0 1
%e 0 0 0 0 0 0 1 0 0 1
%e 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 0 0 0 0 2 1 0 0 0 1
%e 0 0 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 0 0 0 0 0 0 1 0 0 0 0 1
%e 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
%e 0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 1
%e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 1
%e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
%e 0 0 0 0 0 0 0 0 2 0 0 1 0 0 0 0 0 0 0 1
%e Row 12 {0,0,0,0,0,0,2,1,0,0,0,1} corresponds to the factorizations:
%e . . . . . . (3*4) (2*6) . . . (12)
%e (2*2*3)
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t Table[Length[Select[facs[n],Total[#]==k&]],{n,20},{k,n}]
%Y Row sums are A001055. Column sums are A002865.
%Y Cf. A069016, A096276, A301987, A319000, A319005, A319057, A319916.
%K nonn,tabl
%O 1,10
%A _Gus Wiseman_, Oct 22 2018
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