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A300436
Number of odd p-trees of weight n (all proper terminal subtrees have odd weight).
10
1, 1, 1, 2, 2, 5, 5, 12, 13, 35, 37, 98, 107, 304, 336, 927, 1037, 3010, 3367, 9585, 10924, 32126, 36438, 105589, 121045, 359691, 412789, 1211214, 1398168, 4188930, 4831708, 14315544, 16636297, 50079792, 58084208, 173370663, 202101971, 611487744, 712709423
OFFSET
1,4
COMMENTS
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.
FORMULA
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
EXAMPLE
The a(7) = 5 odd p-trees: ((ooo)(ooo)o), (((ooo)oo)oo), ((ooooo)oo), ((ooo)oooo), (ooooooo).
MATHEMATICA
b[n_]:=b[n]=If[n>1, 0, 1]+Sum[Times@@b/@y, {y, Select[IntegerPartitions[n], Length[#]>1&&And@@OddQ/@#&]}];
Table[b[n], {n, 40}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 05 2018
EXTENSIONS
Name corrected by Gus Wiseman, Aug 27 2018
STATUS
approved