A194530 - OEIS

%I #25 Aug 23 2023 08:43:52

%S 0,1,3,9,32,132,623,3314,19628,128126,914005,7074517,59050739,

%T 528741491,5055414317,51406084221,553946196892,6305737560455,

%U 75610546284387,952559077043183,12579235034203780,173759983171005721,2505751777457313815,37657189917162605826

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

%H Alois P. Heinz, <a href="/A194530/b194530.txt">Table of n, a(n) for n = 0..200</a>

%H Soheir Mohamed Khamis, <a href="http://dx.doi.org/10.1007/s11083-011-9213-5">Exact Counting of Unlabeled Rigid Interval Posets Regarding or Disregarding Height</a>, Order (journal) (2011).

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

%F From _Peter Bala_, Aug 21 2023: (Start)

%F Conjectural g.f.:

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

%F 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)

%p w:= proc(t) option remember;

%p       `if`(t=0, 1, expand(convert(series(series(z +z*(subs(

%p            z=z+y+y*z, w(t-1)) -w(t-1)), z, t+1), y, t+1), polynom)))

%p     end:

%p a:= n-> coeff(coeff(w(2+n), z, 2), y, n):

%p seq(a(n), n=0..50);

%t 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_ *)

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

%Y Cf. A138265.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Aug 28 2011