|
%I #8 Aug 26 2018 16:34:52
%S 1,2,4,12,40,136,496,1952,7488,30368,123456,512384,2129664,9068672,
%T 38391552,165642752,713405952,3109135872,13528865792,59591322624,
%U 261549260800,1159547047936,5131968999424,22883893137408,101851069587456,456703499042816,2042949493276672
%N Number of enriched p-trees of weight 2n + 1 in which all outdegrees and all leaves are odd.
%C An enriched p-tree of weight n > 0 is either a single node of weight n, or a finite sequence of at least two enriched p-trees whose weights are weakly decreasing and sum to n.
%H Andrew Howroyd, <a href="/A300652/b300652.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) = (1 - (-1)^n)/2 + Sum_y Product_{i in y} a(i) where the sum is over all non-singleton integer partitions of n with an odd number of parts.
%e The a(3) = 12 trees:
%e 7,
%e (511), (331),
%e ((111)31), (3(111)1), ((311)11), (31111),
%e ((111)(111)1), (((111)11)11), ((11111)11), ((111)1111), (1111111).
%t r[n_]:=r[n]=If[OddQ[n],1,0]+Sum[Times@@r/@y,{y,Select[IntegerPartitions[n],Length[#]>1&&OddQ[Length[#]]&]}];
%t Table[r[n],{n,1,40,2}]
%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^(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 26 2018
%Y Cf. A000009, A000041, A063834, A196545, A273873, A281145, A289501, A298118, A300352, A300353, A300354, A300436, A300439, A300442, A300443, A300574, A300797.
%K nonn
%O 0,2
%A _Gus Wiseman_, Mar 10 2018
|
