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Number of same-trees of weight 2n + 1 in which all outdegrees are odd and all leaves greater than 1.
3

%I #7 Mar 10 2018 19:53:30

%S 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,

%T 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,

%U 3,2,3,34,1,3,1,3,390782,1,1,3,3,3,2198,1,1

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

%C 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.

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

%e The a(13) = 10 odd same-trees with all leaves greater than 1:

%e 27,

%e (999),

%e (99(333)), (9(333)9), ((333)99),

%e (9(333)(333)), ((333)9(333)), ((333)(333)9),

%e ((333)(333)(333)), (333333333).

%t a[n_]:=If[n===1,1,Sum[a[n/d]^d,{d,Select[Rest[Divisors[n]],OddQ]}]];

%t Table[a[n],{n,1,100,2}]

%o (PARI) f(n) = if (n==1, 1, sumdiv(n, d, if ((d > 1) && (d % 2), f(n/d)^d)));

%o a(n) = f(2*n+1); \\ _Michel Marcus_, Mar 10 2018

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

%K nonn

%O 0,5

%A _Gus Wiseman_, Mar 10 2018