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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

Alois P. Heinz, Table of n, a(n) for n = 1..1000

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

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Last modified November 30 19:14 EST 2022. Contains 358453 sequences. (Running on oeis4.)