A217420 Number of rooted unlabeled trees where the root node has degree 2 and both branches are distinct.

0, 0, 0, 1, 2, 6, 14, 37, 92, 239, 613, 1607, 4215, 11185, 29814, 80070, 216061, 586218, 1597292, 4370721, 12003163, 33077327, 91431425, 253454781, 704425087, 1962537755, 5479843060, 15332668869, 42983623237, 120716987723, 339595975795, 956840683968

Offset: 1

Author

Geoffrey Critzer, Oct 19 2012

Keywords

• nonn

References

• F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, page 57.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Charlie Liou and Anthony Mendes, Matrix Representations From Labeled Trees, J. Int. Seq. (2023) Vol. 26, No. 7, Article 23.7.6.

Crossrefs

Cf. A000081 (rooted trees), A000055 (free trees).

Programs

Maple

with(numtheory):
b:= proc(n) option remember; `if`(n<=1, n,
      (add(add(d*b(d), d=divisors(j))*b(n-j), j=1..n-1))/(n-1))
    end:
a:= proc(n) option remember; (add(b(k)*b(n-1-k), k=0..n-1)-
      `if`(irem(n, 2, 'r')=1, b(r), 0))/2
    end:
seq(a(n), n=1..50); #  Alois P. Heinz, May 16 2013

Mathematica

Needs["Combinatorica`"]
nn=30;s[n_,k_]:=s[n,k]=a[n+1-k]+If[n<2k,0,s[n-k,k]];a[1]=1;a[n_]:=a[n]=Sum[a[i]s[n-1,i]i,{i,1,n-1}]/(n-1);rt=Table[a[i],{i,1,nn}];Take[CoefficientList[CycleIndex[AlternatingGroup[2],s]-CycleIndex[SymmetricGroup[2],s]/.Table[s[j]->Table[Sum[rt[[i]]x^(i*k),{i,1,nn}],{k,1,nn}][[j]],{j,1,nn}],x],nn]  (* after code by Robert A. Russell in A000081 *)

Python

# uses function in A000081
def A217420(n): return sum(A000081(i)*A000081(n-1-i) for i in range(1,(n-1)//2+1)) - ((A000081((n-1)//2)+1)*A000081((n-1)//2)//2 if n % 2 else 0) # Chai Wah Wu, Feb 03 2022

Formula

O.g.f.: x * (T(x)^2/2 - T(x^2)/2) where T(x) is o.g.f. for A000081.

a(n) = A000081(n-1) - A000055(n-1) for n > 1.

a(n) = Sum_{1 <= i < j, i + j = m} A000081(i) * A000081(j) + (1 - (-1)^n) * binomial(A000081(m/2),2) / 2 where m = n - 1. - Walt Rorie-Baety, Aug 30 2021