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%I #9 Jan 20 2018 13:16:58
%S 1,0,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0,1,0,1,2,1,0,0,1,0,1,0,0,0,0,
%T 0,1,0,1,3,0,1,0,0,0,1,0,1,0,2,0,0,0,0,0,1,0,1,6,0,0,1,0,0,0,0,1,0,1,
%U 0,0,0,0,0,0,0,0,0,1,0,1,11,4,2,0,1,0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,1,23,0,0,0,0,1,0,0,0,0,0,0,1
%N Regular triangle where T(n,k) is number of k-ary rooted trees with n nodes.
%C Row sums are A298422.
%H Alois P. Heinz, <a href="/A298426/b298426.txt">Rows n = 1..200, flattened</a>
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 1 0 1
%e 0 1 1 0 1
%e 0 1 0 0 0 1
%e 0 1 2 1 0 0 1
%e 0 1 0 0 0 0 0 1
%e 0 1 3 0 1 0 0 0 1
%e 0 1 0 2 0 0 0 0 0 1
%e 0 1 6 0 0 1 0 0 0 0 1
%e 0 1 0 0 0 0 0 0 0 0 0 1
%e 0 1 11 4 2 0 1 0 0 0 0 0 1
%e 0 1 0 0 0 0 0 0 0 0 0 0 0 1
%e 0 1 23 0 0 0 0 1 0 0 0 0 0 0 1
%e 0 1 0 8 0 2 0 0 0 0 0 0 0 0 0 1
%t nn=16;
%t arut[n_,k_]:=If[n===1,{{}},Join@@Function[c,Union[Sort/@Tuples[arut[#,k]&/@c]]]/@Select[IntegerPartitions[n-1],Length[#]===k&]]
%t Table[arut[n,k]//Length,{n,nn},{k,0,n-1}]
%Y Cf. A000005, A000081, A000598, A001190, A001678, A003238, A004111, A067538, A143773, A289078, A289079, A295461, A298118, A298422, A298423, A298424.
%K nonn,tabl
%O 1,24
%A _Gus Wiseman_, Jan 19 2018
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