A307006 Expansion of e.g.f. (2*exp(x)-1-2*x-x^2)/(1-x-x^2).

1, 1, 4, 20, 130, 1052, 10214, 115684, 1497458, 21806372, 352834942, 6279885284, 121932835754, 2564788969108, 58098821674742, 1410088008633812, 36505125340079074, 1004131069129741124, 29244927598399536878, 899066450011962665092, 29094401487631077315482, 988590340245276942963764

Offset: 0

Author

J. Devillet, Mar 19 2019

Keywords

• nonn
• easy

Comments

Number of associative and quasitrivial binary operations on an n-element set that are order-preserving for some total ordering.

Convention a(0) = 1.

Links

• J. Devillet, J.-L. Marichal, and B. Teheux Classifications of quasitrivial semigroups, arXiv:1811.11113 [math.RA], 2018.

Crossrefs

Cf. A000045, A307005.

Programs

Mathematica

Nest[Append[#1, 2 + #2 #1[[-1]] + #2 (#2 - 1) #1[[-2]] ] & @@ {#, Length@ #} &, {1, 1, 4}, 19] (* Michael De Vlieger, Apr 21 2019 *)
With[{nn=30},CoefficientList[Series[(2Exp[x]-1-2x-x^2)/(1-x-x^2),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 12 2020 *)

PARI

my(x='x+O('x^30)); Vec(serlaplace((2*exp(x)-1-2*x-x^2)/(1-x-x^2))) \\ Felix Fröhlich, Mar 19 2019

Formula

Recurrence: a(1) = 1, a(2) = 4, a(n+2) = 2 + (n+2)*a(n+1) + (n+2)*(n+1)*a(n).

a(n) = n!*A000045(n) + 2*Sum_{i=0..n} (n!/(n + 1 - i)!)*A000045(i).

Extensions

More terms from Michel Marcus, Apr 20 2019