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A300355 Number of enriched p-trees of weight n with odd leaves. 7
1, 1, 1, 3, 6, 16, 47, 132, 410, 1254, 4052, 12818, 42783, 139082, 469924, 1563606, 5353966, 18065348, 62491018, 213391790, 743836996, 2565135934, 8994087070, 31251762932, 110245063771, 385443583008, 1365151504722, 4800376128986, 17070221456536, 60289267885410 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

An enriched p-tree of weight n > 0 is either a single node of weight n, or a sequence of two or more enriched p-trees with weakly decreasing weights summing to n.

LINKS

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

FORMULA

O.g.f: (1 + x/(1-x^2) + Prod_{i>0} 1/(1 - a(i)x^i))/2.

a(n) = Sum_{i=1..A000009(n)} A299203(A300351(n,i)).

EXAMPLE

The a(5) = 16 enriched p-trees of weight with odd leaves:

5,

((31)1), ((((11)1)1)1), (((111)1)1), (((11)(11))1), (((11)11)1), ((1111)1),

(3(11)), (((11)1)(11)), ((111)(11)),

(311), (((11)1)11), ((111)11),

((11)(11)1),

((11)111),

(11111).

MATHEMATICA

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

Table[c[n], {n, 30}]

PROG

(PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = n%2 + polcoef(1/prod(k=1, n-1, 1 - v[k]*x^k + O(x*x^n)), n)); concat([1], v)} \\ Andrew Howroyd, Aug 26 2018

CROSSREFS

Cf. A000009, A063834, A078408, A089259, A196545, A279374, A279785, A289501, A294079, A299202, A299203, A300300, A300301, A300352, A300353, A300354.

Sequence in context: A220184 A007002 A305136 * A274294 A201969 A288850

Adjacent sequences:  A300352 A300353 A300354 * A300356 A300357 A300358

KEYWORD

nonn

AUTHOR

Gus Wiseman, Mar 03 2018

STATUS

approved

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Last modified November 12 14:35 EST 2019. Contains 329058 sequences. (Running on oeis4.)