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%I #8 Mar 10 2018 19:53:36
%S 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,
%T 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,
%U 1,3,1656,1,1,3,3,3,43,1,1,31,3,1,3,3,1
%N Number of orderless same-trees of weight 2n + 1 in which all outdegrees are odd and all leaves greater than 1.
%C 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.
%F 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.
%e The a(13) = 6 orderless same-trees: 27, (999), (99(333)), (9(333)(333)), ((333)(333)(333)), (333333333).
%t a[n_]:=If[n===1,1,Sum[Binomial[a[n/d]+d-1,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), binomial(f(n/d)+d-1, 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, A300649.
%K nonn
%O 0,5
%A _Gus Wiseman_, Mar 10 2018
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