OFFSET
0,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..400
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)!/2^i.
G.f.: 1/U(0) where U(k)= 1 - 3*x + x^2 - x*4*k - x^2*(2*k+1)*(2*k+2)/U(k+1) ; (continued fraction, 1-step). - Sergei N. Gladkovskii, Oct 06 2012
G.f.: 1/(1-x)/Q(0), where Q(k)= 1 - x - x*(k+1)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, May 19 2013
a(n) = 2*n*a(n-1) -(2*n-2)*a(n-2) -a(n-3) for n>2. - Alois P. Heinz, Mar 11 2015
a(n) ~ 2^(n + 1/2) * n^n / exp(n-1). - Vaclav Kotesovec, May 05 2024
EXAMPLE
a(2) = 10 = |{ {{},{}}, {{},{1}}, {{},{1,2}}, {{1},{2}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1,2},{3,4}}, {{1,3},{2,4}}, {{1,4},{2,3}} }|.
MAPLE
a:= proc(n) option remember; `if`(n<3, [1, 3, 10][n+1],
2*n*a(n-1)-(2*n-2)*a(n-2)-a(n-3))
end:
seq(a(n), n=0..25); # Alois P. Heinz, Mar 11 2015
MATHEMATICA
Sum[(k+i)!/i!/(k-i)!/2^i, {k, 0, n}, {i, 0, k}]
(* Second program: *)
a[n_] := E*Sqrt[2/Pi]*Sum[BesselK[k + 1/2, 1], {k, 0, n}]; Table[a[n] // Round, {n, 0, 25}] (* Jean-François Alcover, Jul 15 2017 *)
PROG
(PARI) A105748(n) = sum(k=0, n, sum(i=0, k, binomial(k+i, k-i)*binomial(2*i, i)*i!>>i)) \\ M. F. Hasler, Oct 09 2012
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Robert A. Proctor (www.math.unc.edu/Faculty/rap/), Apr 18 2005
STATUS
approved