

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
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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 VinceBona, 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 partitioncounting 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



