OFFSET
0,3
COMMENTS
Number of totally ordered partitions on an n-element set where each non-minimal class contains at most 2 elements.
Convention a(0) = 1.
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..427
Jimmy Devillet, On the single-peakedness property, International summer school "Preferences, decisions and games" (Sorbonne Université, Paris, 2019).
J. Devillet, J.-L. Marichal, and B. Teheux Classifications of quasitrivial semigroups, arXiv:1811.11113 [math.RA], 2018.
FORMULA
Recurrence: a(1) = 1, a(2) = 3, a(n+2) = 1 + (n+2)*a(n+1) + (1/2)*(n+2)*(n+1)*a(n).
a(n) = Sum_{i=0..n} (n!/(n + 1 - i)!)*((sqrt(3)/3)*((1 + sqrt(3))/2)^i - (sqrt(3)/3)*((1 - sqrt(3))/2)^i).
MATHEMATICA
Nest[Append[#1, 1 + #2 #1[[-1]] + #2 (#2 - 1) #1[[-2]]/2 ] & @@ {#, Length@ #} &, {1, 1, 3}, 19] (* Michael De Vlieger, Apr 21 2019 *)
PROG
(PARI) my(x='x+O('x^30)); Vec(serlaplace((2*exp(x)-2*x-x^2)/(2-2*x-x^2))) \\ Felix Fröhlich, Mar 19 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
J. Devillet, Mar 19 2019
EXTENSIONS
More terms from Michel Marcus, Apr 20 2019
STATUS
approved