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A105749 Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a sequence of n sets, each having 1 or 2 elements. 8
1, 2, 14, 222, 6384, 291720, 19445040, 1781750880, 214899027840, 33007837322880, 6290830003852800, 1456812592995513600, 402910665227270323200, 131173228963370155161600, 49656810289225281849907200, 21628258853895305337293568000, 10739534026001485514941587456000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Equivalently, number of sequences of n labeled items such that each item occurs just once or twice. - David Applegate, Dec 08 2008

Also, number of assembly trees for a certain star graph, see Vince-Bona, Theorem 4. - From N. J. A. Sloane, Oct 08 2012

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..100

R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv math.CO.0606404.

Andrew Vince and Miklos Bona, The Number of Ways to Assemble a Graph, arXiv preprint arXiv:1204.3842, 2012.

Index entries for related partition-counting sequences

FORMULA

a(n) = Sum_{k=0..n} C(n,k) * (n+k)! / 2^k.

a(n) = Gamma(n+1)*Hyper2F0([-n, n+1], [], -1/2). - Peter Luschny, Jul 29 2014

a(n) ~ sqrt(Pi) * 2^(n + 1) * n^(2*n + 1/2) / exp(2*n - 1). - Vaclav Kotesovec, Nov 27 2017

EXAMPLE

a(2) = 14 = |{ ({1},{2}), ({2},{1}), ({1},{2,3}), ({2,3},{1}), ({2},{1,3}), ({1,3},{2}), ({3},{1,2}), ({1,2},{3}), ({1,2},{3,4}), ({3,4},{1,2}), ({1,3},{2,4}), ({2,4},{1,3}), ({1,4},{2,3}), ({2,3},{1,4}) }|.

MAPLE

a:= n-> add (binomial(n, k) *(n+k)!/2^k, k=0..n):

seq (a(n), n=0..20);  # Alois P. Heinz, Jul 21 2012

MATHEMATICA

f[n_] := Sum[ Binomial[n, k](n + k)!/2^k, {k, 0, n}]; Table[ f[n], {n, 0, 14}]

CROSSREFS

a(n) = n!*A001515(n). See also A143990.

A003011(n) = Sum[C(n, k)*a(k), 0<=k<=n].

Replace "sets" by "lists": A099022.

Column n=2 of A181731.

Sequence in context: A034405 A197210 A153668 * A251692 A323693 A118086

Adjacent sequences:  A105746 A105747 A105748 * A105750 A105751 A105752

KEYWORD

nonn,easy

AUTHOR

Robert A. Proctor (www.math.unc.edu/Faculty/rap/), Apr 18 2005

EXTENSIONS

More terms from Robert G. Wilson v, Apr 23 2005

STATUS

approved

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Last modified May 19 08:25 EDT 2019. Contains 323389 sequences. (Running on oeis4.)