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.
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
KEYWORD
nonn,easy
AUTHOR
Robert A. Proctor (www.math.unc.edu/Faculty/rap/), Apr 18 2005
STATUS
approved