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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

Table of n, a(n) for n=0..15.

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

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.)