A240159 Triangle read by rows: T(n,k) = number of trees with n vertices and k segments (n >= 2; 1 <= k <= n-1).

1, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 2, 1, 0, 3, 2, 3, 2, 1, 0, 4, 3, 7, 4, 4, 1, 0, 5, 5, 12, 10, 9, 5, 1, 0, 7, 6, 22, 20, 25, 15, 10, 1, 0, 8, 9, 34, 38, 54, 46, 31, 14, 1, 0, 10, 11, 53, 65, 114, 111, 103, 57, 26, 1, 0, 12, 15, 76, 108, 212, 250, 267, 204, 114, 42, 1, 0, 14, 18, 110, 167, 383, 502, 644, 583, 440, 219, 78

Offset: 2

Author

Emeric Deutsch, Apr 02 2014

Keywords

• nonn
• tabl

Comments

A segment of a tree is a path-subtree whose terminal vertices are branching or pendent vertices of the tree. A pendent vertex is a vertex of degree 1; a branching vertex is a vertex of degree >=3.

The author has no formula for obtaining the terms of the sequence. The first Maple program yields the number of segments of a tree identified by the Matula number of any of its associated rooted trees. The second program, making use of the set M of M-indices of the trees with n vertices (given in A235111 for n<=12), yields row n of the triangle. The M-index of a tree is the smallest of the Matula numbers of its associated rooted trees. In the Maple program given below we have n=7.

Examples

			Row 4 is 1, 0, 1. Indeed, there are 2 trees with 4 vertices: the path P_4 having 1 segment and the star S_4 having 3 segments.
Triangle begins:
  1;
  1,0;
  1,0,1;
  1,0,2,1,2;
  1,0,3,2,3,2;
		

Crossrefs

Cf. A000014, A000055, A235111.

Programs

Maple

with(numtheory): seg := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow; n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 and bigomega(pi(n)) = 1 then seg(pi(n)) elif bigomega(n) = 1 and bigomega(pi(n)) = 2 then seg(pi(n))+2 elif bigomega(n) = 1 then seg(pi(n))+1 elif bigomega(r(n)) = 1 and bigomega(s(n)) = 1 then seg(r(n))+seg(s(n))-1 elif bigomega(r(n)) = 1 and bigomega(s(n)) = 2 then seg(r(n))+seg(s(n))+1 elif bigomega(r(n)) = 2 and bigomega(s(n)) = 1 then seg(r(n))+seg(s(n))+1 elif bigomega(r(n)) = 2 and bigomega(s(n)) = 2 then seg(r(n))+seg(s(n))+2 elif bigomega(r(n)) = 1 and 2 < bigomega(s(n)) then seg(r(n))+seg(s(n)) elif 2 < bigomega(r(n)) and bigomega(s(n)) = 1 then seg(r(n))+seg(s(n)) elif bigomega(r(n)) = 2 and 2 < bigomega(s(n)) then seg(r(n))+seg(s(n))+1 elif 2 < bigomega(r(n)) and bigomega(s(n)) = 2 then seg(r(n))+seg(s(n))+1 else seg(r(n))+seg(s(n)) end if end proc:
M := {25, 27, 30, 35, 36, 40, 42, 48, 49, 56, 64}: P := sort(add(x^seg(M[i]), i = 1 .. nops(M))): seq(coeff(P, x, j), j = 1 .. 6);

Mathematica

r[n_] := FactorInteger[n][[1, 1]];
s[n_] := n/r[n];
seg[n_] := Which[n == 1, 0,
  PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == 1, seg[PrimePi[n]],
  PrimeOmega[n] == 1 && PrimeOmega[PrimePi[n]] == 2, seg[PrimePi[n]]+2,
  PrimeOmega[n] == 1, seg[PrimePi[n]]+1,
  PrimeOmega[r[n]] == 1 && PrimeOmega[s[n]] == 1, seg[r[n]] + seg[s[n]]-1,
  PrimeOmega[r[n]] == 1 && PrimeOmega[s[n]] == 2, seg[r[n]] + seg[s[n]]+1,
  PrimeOmega[r[n]] == 2 && PrimeOmega[s[n]] == 1, seg[r[n]] + seg[s[n]]+1,
  PrimeOmega[r[n]] == 2 && PrimeOmega[s[n]] == 2, seg[r[n]] + seg[s[n]]+2,
  PrimeOmega[r[n]] == 1 && 2 < PrimeOmega[s[n]], seg[r[n]] + seg[s[n]],
  2 < PrimeOmega[r[n]] && PrimeOmega[s[n]] == 1, seg[r[n]] + seg[s[n]],
  PrimeOmega[r[n]] == 2 && 2 < PrimeOmega[s[n]], seg[r[n]] + seg[s[n]]+1,
  2 < PrimeOmega[r[n]] && PrimeOmega[s[n]] == 2, seg[r[n]] + seg[s[n]]+1,
  True, seg[r[n]] + seg[s[n]]];
M = {25, 27, 30, 35, 36, 40, 42, 48, 49, 56, 64};
(* M is row[7] in A235111 *)
P = Sort[Sum[x^seg[M[[i]]], {i, 1, Length[M]}]];
Table[Coefficient[P, x, j], {j, 1, 6}] (* Jean-François Alcover, Sep 10 2024, after Maple program *)

Formula

Sum of entries in row n = A000055(n) = number of trees with n vertices.

T(n,n-1) = A000014(n) = number of reduced trees with n vertices.