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.

1, 4, 23, 216, 2937, 52108, 1136591, 29382320, 877838673, 29753600404, 1127881002535, 47278107653768, 2171286661012617, 108417864555606300, 5847857079417024031, 338841578119273846112

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..365

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

Index entries for related partition-counting sequences

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

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