A300355 Number of enriched p-trees of weight n with odd leaves.

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

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: A007002 A305136 A369073 * A360865 A274294 A201969

Adjacent sequences: A300352 A300353 A300354 * A300356 A300357 A300358

Keyword

nonn

Author

Gus Wiseman, Mar 03 2018

Status

approved