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A007453 Number of unlabeled connected series-parallel posets with n nodes.

%I A007453 M2813 #41 Dec 02 2020 14:04:20
%S A007453 1,1,3,9,30,103,375,1400,5380,21073,83950,338878,1383576,5702485,
%T A007453 23696081,99163323,417553252,1767827220,7520966100,32135955585,
%U A007453 137849390424,593407692685,2562695780058,11099806544050,48206136562750,209876865026303,915840095739301
%N A007453 Number of unlabeled connected series-parallel posets with n nodes.
%D A007453 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007453 Alois P. Heinz, <a href="/A007453/b007453.txt">Table of n, a(n) for n = 1..1000</a> (first 100 terms from Jean-François Alcover)
%H A007453 B. I. Bayoumi, M. H. El-Zahar and S. M. Khamis, <a href="https://www.researchgate.net/publication/226027783_Asymptotic_enumeration_ofN-free_partial_orders">Asymptotic enumeration of N-free partial orders</a>, Order 6 (1989), 219-232.
%H A007453 P. J. Cameron, <a href="https://doi.org/10.1016/S0012-365X(96)00226-9">On the probability of connectedness</a>, Discrete Math., 167 (1997), 175-187.
%H A007453 P. J. Cameron, <a href="http://dx.doi.org/10.1016/0012-365X(89)90081-2">Some sequences of integers</a>, Discrete Math., 75 (1989), 89-102.
%H A007453 P. J. Cameron, <a href="http://dx.doi.org/10.1016/S0167-5060(08)70569-7">Some sequences of integers</a>, in "Graph Theory and Combinatorics 1988", ed. B. Bollobas, Annals of Discrete Math., 43 (1989), 89-102.
%H A007453 Soheir M. Khamis, <a href="https://doi.org/10.1016/S0012-365X(03)00106-7">Height counting of unlabeled interval and N-free posets</a>, Discrete Math. 275 (2004), no. 1-3, 165-175.
%H A007453 <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%F A007453 See the 1989 and 1997 papers by Cameron cited above for generating functions, and the 1997 paper for asymptotics.
%F A007453 Inverse Euler transform of A003430. - _Sean A. Irvine_, Jan 04 2018
%F A007453 a(n) = A003430(n) - A007454(n) for n > 1. - _Sean A. Irvine_, Jan 04 2018
%t A007453 terms = 25; A[_] = 1;
%t A007453 Do[A[x_] = Exp[Sum[(1/k)*(A[x^k] + 1/A[x^k] - 2 + x^k), {k, 1, terms+1}]] + O[x]^(terms+1) // Normal, terms+1];
%t A007453 A003430 = CoefficientList[A[x], x] // Rest;
%t A007453 mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
%t A007453 EULERi[b_] := Module[{a, c, i, d}, c = {}; For[i = 1, i <= Length[b], i++, c = Append[c, i*b[[i]] - Sum[c[[d]]*b[[i - d]], {d, 1, i-1}]]]; a = {}; For[i = 1, i <= Length[b], i++, a = Append[a, (1/i)*Sum[mob[i, d]*c[[d]], {d, 1, i}]]]; Return[a]];
%t A007453 EULERi[A003430] (* _Jean-François Alcover_, Jan 23 2020 *)
%Y A007453 Cf. A003430, A007454.
%K A007453 nonn,nice
%O A007453 1,3
%A A007453 _N. J. A. Sloane_, _Mira Bernstein_
%E A007453 Name corrected by _Salah Uddin Mohammad_, Jun 07 2020