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
Keywords
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).
Programs
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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 *)
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)