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Number of odd enriched p-trees of weight n (all outdegrees are odd).
12

%I #9 Aug 26 2018 16:33:19

%S 1,1,2,2,5,7,18,29,75,132,332,651,1580,3268,7961,16966,40709,89851,

%T 215461,484064,1159568,2641812,6337448,14622880,35051341,81609747,

%U 196326305,459909847,1107083238,2611592457,6299122736,14926657167,36069213786,85809507332

%N Number of odd enriched p-trees of weight n (all outdegrees are odd).

%C An odd enriched p-tree of weight n > 0 is either a single node of weight n, or a finite odd-length sequence of at least 3 odd enriched p-trees whose weights are weakly decreasing and sum to n.

%H Andrew Howroyd, <a href="/A300439/b300439.txt">Table of n, a(n) for n = 1..500</a>

%e The a(6) = 7 odd enriched p-trees: 6, (411), (321), (222), ((111)21), ((211)11), (21111).

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

%t Array[f,40]

%o (PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)) - 1/prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)), n)/2); v} \\ _Andrew Howroyd_, Aug 26 2018

%Y Cf. A000009, A027193, A063834, A078408, A196545, A273873, A289501, A294079, A298118, A299202, A299203, A300300, A300301, A300436, A300440.

%K nonn

%O 1,3

%A _Gus Wiseman_, Mar 05 2018