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A331233 Number of unlabeled rooted trees with n vertices and more than two branches of the root. 4
0, 0, 0, 1, 2, 5, 12, 30, 75, 194, 501, 1317, 3485, 9302, 24976, 67500, 183290, 500094, 1369939, 3766831, 10391722, 28756022, 79794407, 221987348, 619019808, 1729924110, 4844242273, 13590663071, 38195831829, 107523305566, 303148601795, 855922155734, 2419923253795 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 500 terms from Andrew Howroyd)

FORMULA

For n > 1, a(n) = Sum_{k > 2} A033185(n - 1, k).

G.f.: f(x) - x*(1 + f(x) + (f(x)^2 + f(x^2))/2) where f(x) is the g.f. of A000081. - Andrew Howroyd, Jan 22 2020

EXAMPLE

The a(4) = 1 through a(7) = 12 rooted trees:

  (ooo)  (oooo)   (ooooo)    (oooooo)

         (oo(o))  (oo(oo))   (oo(ooo))

                  (ooo(o))   (ooo(oo))

                  (o(o)(o))  (oooo(o))

                  (oo((o)))  (o(o)(oo))

                             (oo((oo)))

                             (oo(o)(o))

                             (oo(o(o)))

                             (ooo((o)))

                             ((o)(o)(o))

                             (o(o)((o)))

                             (oo(((o))))

MAPLE

g:= proc(n, i, t) option remember; `if`(n=0, `if`(t=0, 1, 0),

      `if`(i<1, 0, add(binomial(g(i-1$2, 0)+j-1, j)*

         g(n-i*j, i-1, max(0, t-j)), j=0..n/i)))

    end:

a:= n-> g(n-1$2, 3):

seq(a(n), n=1..40);  # Alois P. Heinz, Jan 22 2020

MATHEMATICA

urt[n_]:=Join@@Table[Union[Sort/@Tuples[urt/@ptn]], {ptn, IntegerPartitions[n-1]}];

Table[Length[Select[urt[n], Length[#]>2&]], {n, 10}]

(* Second program: *)

g[n_, i_, t_] := g[n, i, t] = If[n == 0, If[t == 0, 1, 0],

     If[i < 1, 0, Sum[Binomial[g[i - 1, i - 1, 0] + j - 1, j]*

     g[n - i*j, i - 1, Max[0, t - j]], {j, 0, n/i}]]];

a[n_] := g[n-1, n-1, 3];

Array[a, 40] (* Jean-François Alcover, May 20 2021, after Alois P. Heinz *)

PROG

(PARI) \\ TreeGf gives gf of A000081.

TreeGf(N)={my(A=vector(N, j, 1)); for (n=1, N-1, A[n+1] = 1/n * sum(k=1, n, sumdiv(k, d, d*A[d]) * A[n-k+1] ) ); x*Ser(A)}

seq(n)={my(g=TreeGf(n)); Vec(g - x*(1 + g + (g^2 + subst(g, x, x^2))/2), -n)} \\ Andrew Howroyd, Jan 22 2020

CROSSREFS

The Matula-Goebel numbers of these trees are given by A033942.

The series-reduced case is A331488.

The lone-child-avoiding case is (also) A331488.

The labeled version is A331577.

Unlabeled rooted trees are counted by A000081.

Cf. A001678, A001679, A004111, A033185, A060313, A206429, A331490, A331578.

Sequence in context: A326793 A026580 A092247 * A108360 A051163 A051450

Adjacent sequences:  A331230 A331231 A331232 * A331234 A331235 A331236

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jan 21 2020

STATUS

approved

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Last modified September 23 20:42 EDT 2021. Contains 347617 sequences. (Running on oeis4.)