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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. 4
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
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
a(n) ~ 2^(2*n + 1/2) * n^n / exp(n - 1/2). - Vaclav Kotesovec, May 15 2022
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
Table[Sum[(k+i)!/i!/(k-i)!, {k, 0, n}, {i, 0, k}], {n, 0, 20}]
CROSSREFS
First differences: A001517.
Replace "collection" by "sequence": A082765.
Replace "lists" by "sets": A105748.
Sequence in context: A221371 A056785 A188404 * A099692 A283499 A305787
KEYWORD
nonn,easy
AUTHOR
Robert A. Proctor (www.math.unc.edu/Faculty/rap/), Apr 18 2005
STATUS
approved

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Last modified March 29 10:22 EDT 2024. Contains 371268 sequences. (Running on oeis4.)