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

Author

Gus Wiseman, Sep 13 2016

Keywords

• nonn

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).

Examples

			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)}.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..681
• Gus Wiseman, Comcategories and Multiorders, (pdf version)

Crossrefs

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

Programs

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]