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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. 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 December 6 21:45 EST 2019. Contains 329809 sequences. (Running on oeis4.)