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A318485 Number of p-trees of weight 2n + 1 in which all outdegrees are odd.

%I A318485 #9 Aug 31 2018 15:24:41
%S A318485 1,1,2,5,13,37,107,336,1037,3367,10924,36438,121045,412789,1398168,
%T A318485 4831708,16636297,58084208,202101971,712709423,2502000811,8880033929,
%U A318485 31428410158,112199775788,399383181020,1433385148187,5128572792587,18481258241133
%N A318485 Number of p-trees of weight 2n + 1 in which all outdegrees are odd.
%C A318485 A p-tree of weight n with odd outdegrees is either a single node (if n = 1) or a finite odd-length sequence of at least 3 p-trees with odd outdegrees whose weights are weakly decreasing and sum to n.
%H A318485 Andrew Howroyd, <a href="/A318485/b318485.txt">Table of n, a(n) for n = 0..500</a>
%e A318485 The a(4) = 13 p-trees of weight 9 with odd outdegrees:
%e A318485   ((((ooo)oo)oo)oo)
%e A318485   (((ooo)(ooo)o)oo)
%e A318485   (((ooo)oo)(ooo)o)
%e A318485   ((ooo)(ooo)(ooo))
%e A318485   (((ooooo)oo)oo)
%e A318485   (((ooo)oooo)oo)
%e A318485   ((ooooo)(ooo)o)
%e A318485   (((ooo)oo)oooo)
%e A318485   ((ooo)(ooo)ooo)
%e A318485   ((ooooooo)oo)
%e A318485   ((ooooo)oooo)
%e A318485   ((ooo)oooooo)
%e A318485   (ooooooooo)
%t A318485 b[n_]:=b[n]=If[n>1,0,1]+Sum[Times@@b/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
%t A318485 Table[b[n],{n,1,20,2}]
%o A318485 (PARI) seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, v[n] = polcoef(1/prod(k=1, n-1, 1 - v[k]*x^(2*k-1) + O(x^(2*n))) - 1/prod(k=1, n-1, 1 + v[k]*x^(2*k-1) + O(x^(2*n))), 2*n-1)/2); v} \\ _Andrew Howroyd_, Aug 27 2018
%Y A318485 Cf. A027193, A063834, A078408, A196545, A279374, A289501, A298118, A300300, A300301, A300355, A300436, A300647, A300652, A300797, A302243.
%K A318485 nonn
%O A318485 0,3
%A A318485 _Gus Wiseman_, Aug 27 2018