|
%I #5 Dec 01 2022 08:55:56
%S 0,1,2,0,3,1,1,-1,1,2,4,0,2,0,2,-2,2,0,0,1,0,3,2,-1,2,1,0,-1,3,1,5,-3,
%T 3,1,1,-1,1,-1,1,0,3,-1,1,2,1,1,3,-2,-1,1,1,0,-1,-1,3,-2,-1,2,3,0,1,4,
%U -1,-4,1,2,1,0,1,0,2,-2,1,0,1,-2,2,0,4,-1
%N Difference between the node-height and the number of leaves in the rooted tree with Matula-Goebel number n.
%C Node-height is the number of nodes in the longest path from root to leaf.
%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) = A358552(n) - A109129(n).
%e The tree (oo(oo(o))) with Matula-Goebel number 148 has node-height 4 and 5 leaves, so a(148) = -1.
%t MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Table[(Depth[MGTree[n]]-1)-Count[MGTree[n],{},{0,Infinity}],{n,1000}]
%Y Positions of first appearances are A007097 and latter terms of A000079.
%Y Positions of 0's are A358577.
%Y Other differences: A358580, A358724, A358729.
%Y A000081 counts rooted trees, ordered A000108.
%Y A034781 counts rooted trees by nodes and height, ordered A080936.
%Y A055277 counts rooted trees by nodes and leaves, ordered A001263.
%Y MG statistics: A061775, A109082, A109129, A196050, A342507, A358552.
%Y MG core: A000040, A000720, A001222, A056239, A112798.
%Y Cf. A185650, A206487, A209638, A358576, A358578, A358587, A358730.
%K sign
%O 1,3
%A _Gus Wiseman_, Nov 29 2022
|
