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A319369 Number of series-reduced rooted trees with n leaves of n colors. 4
1, 3, 28, 430, 9376, 269675, 9632960, 411395268, 20445999734, 1159248404721, 73846864163348, 5221802726902476, 405858598184643930, 34392275731729465799, 3155760058245300968416, 311720334688779807141832, 32980137195294216968253900, 3720954854814866649904474180 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

Not all of the n colors need to be used.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 1..340

V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.

FORMULA

a(n) ~ c * d^n * n^(n - 3/2), where d = 2.588699449562089830805384431942090... and c = 0.2580000331300831455241033648... - Vaclav Kotesovec, Sep 18 2019

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-> A(n$2):

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

MATHEMATICA

b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[A[i, k] + j - 1, j]*b[n - i*j, i - 1, k], {j, 0, n/i}]]];

A[n_, k_] := If[n < 2, n*k, b[n, n - 1, k]];

a[n_] := A[n, n];

a /@ Range[1, 20] (* Jean-Fran├žois Alcover, Sep 24 2019, after Alois P. Heinz *)

PROG

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

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

CROSSREFS

Main diagonal of A319254.

Cf. A000311 (1 leaf of each color), A316651.

Sequence in context: A298696 A143636 A219532 * A210854 A108288 A060545

Adjacent sequences:  A319366 A319367 A319368 * A319370 A319371 A319372

KEYWORD

nonn

AUTHOR

Andrew Howroyd, Sep 17 2018

STATUS

approved

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Last modified December 15 17:03 EST 2019. Contains 330000 sequences. (Running on oeis4.)