A300436 Number of odd p-trees of weight n (all proper terminal subtrees have odd weight).

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.

Links

Table of n, a(n) for n=1..39.

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}]

Crossrefs

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

Sequence in context: A135153 A097896 A030223 * A056504 A122205 A368923

Adjacent sequences: A300433 A300434 A300435 * A300437 A300438 A300439

Keyword

nonn

Author

Gus Wiseman, Mar 05 2018

Extensions

Name corrected by Gus Wiseman, Aug 27 2018

Status

approved