A301343 Regular triangle where T(n,k) is the number of planted achiral (or generalized Bethe) trees with n nodes and k leaves.

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

Offset: 1

Author

Gus Wiseman, Mar 19 2018

Keywords

• nonn
• tabl

Examples

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

Crossrefs

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

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

Programs

Mathematica

tri[n_,k_]:=If[k===1,1,If[k>=n,0,Sum[tri[n-k,d],{d,Divisors[k]}]]];
Table[tri[n,k],{n,10},{k,n}]

Formula

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