A300650 Number of orderless same-trees of weight 2n + 1 in which all outdegrees are odd and all leaves greater than 1.

1, 1, 1, 1, 2, 1, 1, 3, 1, 1, 3, 1, 2, 6, 1, 1, 3, 3, 1, 3, 1, 1, 19, 1, 2, 3, 1, 3, 3, 1, 1, 21, 3, 1, 3, 1, 1, 28, 3, 1, 68, 1, 3, 3, 1, 3, 3, 3, 1, 25, 1, 1, 71, 1, 1, 3, 1, 3, 27, 3, 2, 3, 8, 1, 3, 1, 3, 1656, 1, 1, 3, 3, 3, 43, 1, 1, 31, 3, 1, 3, 3, 1

Offset: 0

Author

Gus Wiseman, Mar 10 2018

Keywords

• nonn

Comments

An orderless same-tree of weight n > 0 is either a single node of weight n, or a finite multiset of two or more orderless same-trees whose weights are all equal and sum to n.

Examples

			The a(13) = 6 orderless same-trees: 27, (999), (99(333)), (9(333)(333)), ((333)(333)(333)), (333333333).
		

Crossrefs

Cf. A003238, A006241, A063834, A069283, A273873, A281145, A289078, A289079, A289501, A298118, A300436, A300439, A300574, A300647, A300648, A300649.

Programs

Mathematica

a[n_]:=If[n===1,1,Sum[Binomial[a[n/d]+d-1,d],{d,Select[Rest[Divisors[n]],OddQ]}]];
Table[a[n],{n,1,100,2}]

PARI

f(n) = if (n==1, 1, sumdiv(n, d, if ((d > 1) && (d % 2), binomial(f(n/d)+d-1, d))));
a(n) = f(2*n+1); \\ Michel Marcus, Mar 10 2018

Formula

a(1) = 1; a(n > 1) = Sum_d binomial(a(n/d) + d - 1, d) where the sum is over odd divisors of n greater than 1.