A276687 Number of prime plane trees of weight prime(n).

1, 1, 2, 4, 11, 30, 122, 336, 1412, 15129, 44561, 417542, 2479120, 7540843, 35983502, 451454834, 5313515136, 16809858904, 190077477328, 1124302066470, 3521811953565, 38563707677633, 240966297786218, 3192420711942298, 95433674596402663, 567734580765228356

Offset

1,3

Comments

A prime plane tree is either (case 1) a prime number, or (case 2) a sequence of prime plane trees whose weights are an integer partition of a prime number, where the weight of a tree is the sum of weights of its branches. Prime plane trees are "multichains" in the multiorder of integer partitions of prime numbers into prime parts (A056768).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..681

Gus Wiseman, Comcategories and Multiorders, (pdf version)

Example

The a(5) = 11 prime plane trees of weight A000040(5) = 11 are: {11, (3,3,5), (3,3,(2,3)), (2,2,7), (2,2,(2,5)), (2,2,(2,(2,3))), (2,2,(2,2,3)), (2,3,3,3), (2,2,2,5), (2,2,2,(2,3)), (2,2,2,2,3)}.

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i=2, 0,
       b(n, prevprime(i)))+`if`(i>n, 0, b(n-i, i)*(1+
      `if`(i>2, b(i, prevprime(i)), 0))))
    end:
a:= n-> `if`(n<3, 1, 1+b(ithprime(n), ithprime(n-1))):
seq(a(n), n=1..40);  # Alois P. Heinz, Sep 15 2016

Mathematica

n=20;
ser=Product[1/(1-c[Prime[i]]*x^Prime[i]), {i, 1, n}];
sys=Table[c[Prime[i]]==Expand[SeriesCoefficient[ser, {x, 0, Prime[i]}]-c[Prime[i]]+1], {i, 1, n}];
Block[{c}, Set@@@sys]

Crossrefs

Cf. A000040, A056768, A000607, A100118, A196545, A273873.

Sequence in context: A102814 A193059 A034770 * A298891 A002387 A325922

Adjacent sequences: A276684 A276685 A276686 * A276688 A276689 A276690

Keyword

nonn

Author

Gus Wiseman, Sep 13 2016

Status

approved