A300649 Number of 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, 10, 1, 1, 3, 3, 1, 3, 1, 1, 62, 1, 2, 3, 1, 3, 3, 1, 1, 158, 3, 1, 3, 1, 1, 254, 3, 1, 1514, 1, 3, 3, 1, 3, 3, 3, 1, 2078, 1, 1, 2461, 1, 1, 3, 1, 3, 8222, 3, 2, 3, 34, 1, 3, 1, 3, 390782, 1, 1, 3, 3, 3, 2198, 1, 1

Offset: 0

Author

Gus Wiseman, Mar 10 2018

Keywords

• nonn

Comments

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

Examples

			The a(13) = 10 odd same-trees with all leaves greater than 1:
27,
(999),
(99(333)), (9(333)9), ((333)99),
(9(333)(333)), ((333)9(333)), ((333)(333)9),
((333)(333)(333)), (333333333).
		

Crossrefs

Cf. A003238, A006241, A063834, A069283, A273873, A281145, A289078, A289079, A289501, A298118, A300436, A300439, A300574, A300647, A300648, A300650.

Programs

Mathematica

a[n_]:=If[n===1,1,Sum[a[n/d]^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), f(n/d)^d)));
a(n) = f(2*n+1); \\ Michel Marcus, Mar 10 2018

Formula

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