A358580 Difference between the number of leaves and the number of internal (non-leaf) nodes in the rooted tree with Matula-Goebel number n.

1, 0, -1, 1, -2, 0, 0, 2, -1, -1, -3, 1, -1, 1, -2, 3, -1, 0, 1, 0, 0, -2, -2, 2, -3, 0, -1, 2, -2, -1, -4, 4, -3, 0, -1, 1, 0, 2, -1, 1, -2, 1, 0, -1, -2, -1, -3, 3, 1, -2, -1, 1, 2, 0, -4, 3, 1, -1, -2, 0, -1, -3, 0, 5, -2, -2, 0, 1, -2, 0, -1, 2, -1, 1, -3

Offset: 1

Author

Gus Wiseman, Nov 25 2022

Keywords

• sign

Comments

The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.

Examples

			The Matula-Goebel number of ((ooo(o))) is 89, and it has 4 leaves and 3 internal nodes, so a(89) = 1.
		

Crossrefs

Zeros are A358578, counted by A185650 (ordered A358579).

Positions of positive terms are counted by A358581, negative A358582.

Positions of nonnegative terms are counted by A358583, nonpositive A358584.

A000081 counts rooted trees, ordered A000108.

A034781 counts trees by nodes and height.

A055277 counts trees by nodes and leaves, ordered A001263.

MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.

Cf. A000040, A000720, A001222, A007097, A056239, A112798.

Cf. A206487, A358371, A358576, A358577, A358592.

Programs

Mathematica

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Count[MGTree[n],{},{0,Infinity}]-Count[MGTree[n],[_],{0,Infinity}],{n,100}]

Formula

a(n) = A109129(n) - A342507(n).