%I #14 Sep 04 2018 09:40:13

%S 1,1,1,2,2,5,5,12,13,35,37,98,107,304,336,927,1037,3010,3367,9585,

%T 10924,32126,36438,105589,121045,359691,412789,1211214,1398168,

%U 4188930,4831708,14315544,16636297,50079792,58084208,173370663,202101971,611487744,712709423

%N Number of odd p-trees of weight n (all proper terminal subtrees have odd weight).

%C An odd p-tree of weight n > 0 is either a single node (if n = 1) or a finite sequence of at least 3 odd p-trees whose weights are weakly decreasing odd numbers summing to n.

%F O.g.f: x + Product_{n odd} 1/(1 - a(n)*x^n) - Sum_{n odd} a(n)*x^n. - _Gus Wiseman_, Aug 27 2018

%e The a(7) = 5 odd p-trees: ((ooo)(ooo)o), (((ooo)oo)oo), ((ooooo)oo), ((ooo)oooo), (ooooooo).

%t b[n_]:=b[n]=If[n>1,0,1]+Sum[Times@@b/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&And@@OddQ/@#&]}];

%t Table[b[n],{n,40}]

%Y Cf. A000009, A027193, A063834, A078408, A196545, A279374, A279785, A289501, A298118, A299202, A299203, A300300, A300301, A300355, A300439, A300440.

%K nonn

%O 1,4

%A _Gus Wiseman_, Mar 05 2018

%E Name corrected by _Gus Wiseman_, Aug 27 2018