A358580 - OEIS

%I #4 Nov 26 2022 12:35:32

%S 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,

%T -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,

%U 3,1,-1,-2,0,-1,-3,0,5,-2,-2,0,1,-2,0,-1,2,-1,1,-3

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

%C 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.

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

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

%t MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t Table[Count[MGTree[n],{},{0,Infinity}]-Count[MGTree[n],_[__],{0,Infinity}],{n,100}]

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

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

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

%Y A000081 counts rooted trees, ordered A000108.

%Y A034781 counts trees by nodes and height.

%Y A055277 counts trees by nodes and leaves, ordered A001263.

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

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

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

%K sign

%O 1,5

%A _Gus Wiseman_, Nov 25 2022