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

1,5

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.

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

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A339985 A026598 A006864 * A071636 A263758 A100067

Adjacent sequences: A217417 A217418 A217419 * A217421 A217422 A217423

Keyword

nonn

Author

Geoffrey Critzer, Oct 19 2012

Status

approved