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Regular triangle where T(n,k) is the number of planted achiral (or generalized Bethe) trees with n nodes and k leaves.
7

%I #5 Mar 19 2018 22:06:15

%S 1,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,2,1,1,1,0,1,3,2,2,1,1,0,1,3,2,2,1,1,

%T 1,0,1,4,2,4,1,2,1,1,0,1,4,3,4,1,3,1,1,1,0,1,5,3,6,2,4,1,2,1,1,0,1,5,

%U 3,6,2,4,1,2,1,1,1,0,1,6,4,9,2,7,1,4,2,2,1,1,0

%N Regular triangle where T(n,k) is the number of planted achiral (or generalized Bethe) trees with n nodes and k leaves.

%F T(n,1) = 1, T(n,k) = 0 if n <= k, otherwise T(n,k) = Sum_{d|k} T(n - k, d).

%e Triangle begins:

%e 1

%e 1 0

%e 1 1 0

%e 1 1 1 0

%e 1 2 1 1 0

%e 1 2 1 1 1 0

%e 1 3 2 2 1 1 0

%e 1 3 2 2 1 1 1 0

%e 1 4 2 4 1 2 1 1 0

%e 1 4 3 4 1 3 1 1 1 0

%e 1 5 3 6 2 4 1 2 1 1 0

%e The T(9,4) = 4 planted achiral trees: (((((oooo))))), ((((oo)(oo)))), (((oo))((oo))), ((o)(o)(o)(o)).

%t tri[n_,k_]:=If[k===1,1,If[k>=n,0,Sum[tri[n-k,d],{d,Divisors[k]}]]];

%t Table[tri[n,k],{n,10},{k,n}]

%Y Row sums are A003238. A version without the zeroes or first row is A214575.

%Y Cf. A000081, A003238, A004111, A032305, A055277, A214577, A273873, A289078, A289079, A290689, A291443, A298422, A298426, A301342, A301344, A301345.

%K nonn,tabl

%O 1,12

%A _Gus Wiseman_, Mar 19 2018