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A217420 Number of rooted unlabeled trees where the root node has degree 2 and both branches are distinct. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,5
REFERENCES
F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, page 57.
LINKS
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
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Oct 19 2012
STATUS
approved

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Last modified June 22 09:48 EDT 2024. Contains 373568 sequences. (Running on oeis4.)