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A300652 Number of enriched p-trees of weight 2n + 1 in which all outdegrees and all leaves are odd. 3
1, 2, 4, 12, 40, 136, 496, 1952, 7488, 30368, 123456, 512384, 2129664, 9068672, 38391552, 165642752, 713405952, 3109135872, 13528865792, 59591322624, 261549260800, 1159547047936, 5131968999424, 22883893137408, 101851069587456, 456703499042816, 2042949493276672 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..500

FORMULA

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.

EXAMPLE

The a(3) = 12 trees:

7,

(511), (331),

((111)31), (3(111)1), ((311)11), (31111),

((111)(111)1), (((111)11)11), ((11111)11), ((111)1111), (1111111).

MATHEMATICA

r[n_]:=r[n]=If[OddQ[n], 1, 0]+Sum[Times@@r/@y, {y, Select[IntegerPartitions[n], Length[#]>1&&OddQ[Length[#]]&]}];

Table[r[n], {n, 1, 40, 2}]

PROG

(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

CROSSREFS

Cf. A000009, A000041, A063834, A196545, A273873, A281145, A289501, A298118, A300352, A300353, A300354, A300436, A300439, A300442, A300443, A300574, A300797.

Sequence in context: A214761 A327845 A056236 * A028329 A204678 A025227

Adjacent sequences:  A300649 A300650 A300651 * A300653 A300654 A300655

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 10 2018

STATUS

approved

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Last modified June 20 06:23 EDT 2021. Contains 345157 sequences. (Running on oeis4.)