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

1,5

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.

Links

Table of n, a(n) for n=1..75.

Formula

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

Example

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

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}]

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.

Sequence in context: A280912 A332662 A364036 * A029303 A361163 A361500

Adjacent sequences: A358577 A358578 A358579 * A358581 A358582 A358583

Keyword

sign

Author

Gus Wiseman, Nov 25 2022

Status

approved