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A316651 Number of series-reduced rooted trees with n leaves spanning an initial interval of positive integers. 29
1, 2, 12, 112, 1444, 24086, 492284, 11910790, 332827136, 10546558146, 373661603588, 14636326974270, 628032444609396, 29296137817622902, 1476092246351259964, 79889766016415899270, 4622371378514020301740, 284719443038735430679268, 18601385258191195218790756 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

A rooted tree is series-reduced if every non-leaf node has at least two branches.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 1..200

FORMULA

From Vaclav Kotesovec, Sep 18 2019: (Start)

a(n) ~ c * d^n * n^(n-1), where d = 1.37392076830840090205551979... and c = 0.41435722857311602982846...

a(n) ~ 2*log(2)*A326396(n)/n. (End)

EXAMPLE

The a(3) = 12 trees:

  (1(11)), (111),

  (1(12)), (2(11)), (112),

  (1(22)), (2(12)), (122),

  (1(23)), (2(13)), (3(12)), (123).

MAPLE

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add(binomial(A(i, k)+j-1, j)*b(n-i*j, i-1, k), j=0..n/i)))

    end:

A:= (n, k)-> `if`(n<2, n*k, b(n, n-1, k)):

a:= n-> add(add(A(n, k-j)*(-1)^j*binomial(k, j), j=0..k-1), k=1..n):

seq(a(n), n=1..20);  # Alois P. Heinz, Sep 18 2018

MATHEMATICA

sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];

mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

gro[m_]:=If[Length[m]==1, m, Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m], Length[#]>1&])]];

allnorm[n_Integer]:=Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1];

Table[Sum[Length[gro[m]], {m, allnorm[n]}], {n, 5}]

PROG

(PARI) \\ here R(n, k) is A000669, A050381, A220823, ...

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

R(n, k)={my(v=[k]); for(n=2, n, v=concat(v, EulerT(concat(v, [0]))[n])); v}

seq(n)={sum(k=1, n, R(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 14 2018

CROSSREFS

Cf. A000081, A000311, A000669, A001678, A005804, A034691, A141268, A292504.

Cf. A316652, A316653, A316654, A316655, A316656, A319369.

Row sums of A319376.

Sequence in context: A214225 A185190 A227460 * A330654 A091481 A053312

Adjacent sequences:  A316648 A316649 A316650 * A316652 A316653 A316654

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 09 2018

EXTENSIONS

Terms a(9) and beyond from Andrew Howroyd, Sep 14 2018

STATUS

approved

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Last modified February 18 15:30 EST 2020. Contains 332019 sequences. (Running on oeis4.)