A307005 Expansion of e.g.f. (2*exp(x)-2*x-x^2)/(2-2*x-x^2).

1, 1, 3, 13, 71, 486, 3982, 38081, 416145, 5116222, 69888746, 1050168417, 17214678241, 305703953660, 5846391071172, 119794781201881, 2618283427770737, 60802908515558346, 1495049717728972990, 38803241993010963977, 1060124286228724147641, 30411290829335509535632

Offset: 0

Author

J. Devillet, Mar 19 2019

Keywords

• nonn
• easy

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.

Crossrefs

Cf. A307006.

Programs

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 *)

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

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).

Extensions

More terms from Michel Marcus, Apr 20 2019