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Number of odd strict trees of weight n (all outdegrees are odd).
5

%I #9 Aug 26 2018 02:35:03

%S 1,1,1,1,1,2,2,3,5,7,11,18,27,45,75,125,207,353,591,1013,1731,2984,

%T 5122,8905,15369,26839,46732,81850,142932,251693,441062,778730,

%U 1370591,2425823,4281620,7601359,13447298,23919512,42444497,75632126,134454505,240100289

%N Number of odd strict trees of weight n (all outdegrees are odd).

%C An odd strict tree of weight n is either a single node of weight n, or a finite odd-length sequence of at least 3 odd strict trees with strictly decreasing weights summing to n.

%H Andrew Howroyd, <a href="/A300440/b300440.txt">Table of n, a(n) for n = 1..500</a>

%e The a(10) = 7 odd strict trees: 10, (721), (631), (541), (532), ((421)21), ((321)31).

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

%t Array[g,20]

%o (PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(prod(k=1, n-1, 1 + v[k]*x^k + O(x*x^n)) - prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)/2); v} \\ _Andrew Howroyd_, Aug 25 2018

%Y Cf. A000009, A000992, A032305, A063834, A078408, A089259, A196545, A273873, A279785, A289501, A298118, A300301, A300352, A300353, A300436, A300439.

%K nonn

%O 1,6

%A _Gus Wiseman_, Mar 05 2018