login
Regular triangle where T(n,k) is number of k-ary rooted trees with n nodes.
18

%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