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A339644 Number of rooted trees on n nodes with labels covering an initial interval of positive integers. 2
1, 3, 21, 214, 3004, 53696, 1169220, 30017582, 887835091, 29728120594, 1111619802614, 45914106227815, 2076062017348677, 101996651482313080, 5410363994433018486, 308174409706787225523, 18760485689929220881741, 1215547422537201878074293, 83520534389622385511232635 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

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

EXAMPLE

The a(3) = 21 rooted trees are:

  (1(11)), (1(1(1))), (1(12)), (1(22)), (1(1(2))), (1(2(1))), (1(2(2))), (2(12)), (2(11)), (2(2(1))), (2(1(2))), (2(1(1))), (1(23)), (1(2(3))), (1(3(2))), (2(13)), (2(1(3))), (2(3(1))), (3(12)), (3(1(2))), (3(2(1))).

MAPLE

b:= proc(n, k) option remember; `if`(n<2, k*n, (add(add(b(d, k)*

      d, d=numtheory[divisors](j))*b(n-j, k), j=1..n-1))/(n-1))

    end:

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

seq(a(n), n=1..21);  # Alois P. Heinz, Dec 11 2020

MATHEMATICA

b[n_, k_] := b[n, k] = If[n<2, k*n, (Sum[Sum[b[d, k]*d, {d, Divisors[j]}]* b[n - j, k], {j, 1, n - 1}])/(n - 1)];

a[n_] := Sum[Sum[b[n, k - j]*Binomial[k, j]*(-1)^j, {j, 0, k}], {k, 0, n}];

Array[a, 21] (* Jean-Fran├žois Alcover, Jan 04 2021, after Alois P. Heinz *)

PROG

(PARI) \\ See A141610 for U(n, k).

seq(n)={sum(k=1, n, U(n, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)))}

CROSSREFS

Row sums of A141610.

Sequence in context: A087918 A088926 A291743 * A120972 A168479 A158838

Adjacent sequences:  A339641 A339642 A339643 * A339645 A339646 A339647

KEYWORD

nonn,changed

AUTHOR

Andrew Howroyd, Dec 11 2020

STATUS

approved

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Last modified January 17 22:55 EST 2021. Contains 340247 sequences. (Running on oeis4.)