A194530 Number of unlabeled rigid interval posets with n non-maximal and 2 maximal elements.

0, 1, 3, 9, 32, 132, 623, 3314, 19628, 128126, 914005, 7074517, 59050739, 528741491, 5055414317, 51406084221, 553946196892, 6305737560455, 75610546284387, 952559077043183, 12579235034203780, 173759983171005721, 2505751777457313815, 37657189917162605826

Offset

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..200

Soheir Mohamed Khamis, Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height, Order (journal) (2011).

Formula

a(n) = [ y^n z^2 ] W(y,z); W(y,z) = z + z*(W(y,y+z+yz) - W(y,z)).

From Peter Bala, Aug 21 2023: (Start)

Conjectural g.f.:

1) A(x) = Sum_{n >= 0} n*(Product_{i = 1..n} 1 - 1/(1+x)^i).

2) A(x) = (1/2)*Sum_{n >= 0} n*(n+1)/(1+x)^(n+1) * (Product_{i = 1..n} 1 - 1/(1+x)^i). Cf. A138265. (End)

Maple

w:= proc(t) option remember;
      `if`(t=0, 1, expand(convert(series(series(z +z*(subs(
           z=z+y+y*z, w(t-1)) -w(t-1)), z, t+1), y, t+1), polynom)))
    end:
a:= n-> coeff(coeff(w(2+n), z, 2), y, n):
seq(a(n), n=0..50);

Mathematica

w[t_] := w[t] = If[t == 0, 1, Expand[Normal[Series[Series[z+z*((w[t-1] /. z -> z+y+y*z)-w[t-1]), {z, 0, t+1}], {y, 0, t+1}]]]]; a[n_] := a[n] = Coefficient[Coefficient[w[2+n], z, 2], y, n]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 0, 50}] (* Jean-François Alcover, Mar 05 2014, after Alois P. Heinz *)

Crossrefs

2nd column of A193344, Column k=2 of A218757.

Cf. A138265.

Sequence in context: A320180 A183425 A039628 * A324238 A005964 A246138

Adjacent sequences: A194527 A194528 A194529 * A194531 A194532 A194533

Keyword

nonn

Author

Alois P. Heinz, Aug 28 2011

Status

approved