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 A105747 Number of ways to use the elements of {1,..,k}, 0<=k<=2n, once each to form a collection of n (possibly empty) lists, each of length at most 2. 3
 1, 4, 23, 216, 2937, 52108, 1136591, 29382320, 877838673, 29753600404, 1127881002535, 47278107653768, 2171286661012617, 108417864555606300, 5847857079417024031, 338841578119273846112 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 LINKS R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions! arXiv math.CO.0606404. FORMULA a(n) = sum(0<=i<=k<=n, (k+i)!/i!/(k-i)! ). a(n+3) = (4*n+11)*a(n+2) - (4*n+9)*a(n+1) - a(n) - Benoit Cloitre, May 26 2006 G.f.: 1/(1-x)/Q(0), where Q(k)= 1 - x - 2*x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 17 2013 EXAMPLE a(2)=23: {(),()}, {(),(1)}, {(),(1,2)}, {(),(2,1)}, {(1),(2)}, {(1),(2,3)}, {(1),(3,2)}, ..., {(1,4),(2,3)}, {(1,4),(3,2)}, {(4,1),(2,3)}, {(4,1),(3,2)}. MATHEMATICA Sum[(k+i)!/i!/(k-i)!, {k, 0, n}, {i, 0, k}] CROSSREFS First differences: A001517. Replace "collection" by "sequence": A082765. Replace "lists" by "sets": A105748. Sequence in context: A221371 A056785 A188404 * A099692 A283499 A305787 Adjacent sequences:  A105744 A105745 A105746 * A105748 A105749 A105750 KEYWORD nonn,easy AUTHOR Robert A. Proctor (www.math.unc.edu/Faculty/rap/), Apr 18 2005 STATUS approved

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Last modified December 9 22:27 EST 2019. Contains 329880 sequences. (Running on oeis4.)