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A007454 Number of unlabeled disconnected series-parallel posets with n nodes.

%I A007454 M1630 #37 Jan 18 2022 19:34:32
%S A007454 1,1,2,6,18,64,227,856,3280,12885,51342,207544,847886,3497384,
%T A007454 14541132,60884173,256480895,1086310549,4623128656,19759964149,
%U A007454 84784735379,365066645854,1576927900803,6831518134251,29674505668536,129216630647787,563949605921815
%N A007454 Number of unlabeled disconnected series-parallel posets with n nodes.
%C A007454 a(1) = 0 would make more sense, but original article has a(1) = 1. - _Sean A. Irvine_, Jan 04 2018
%D A007454 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A007454 Alois P. Heinz, <a href="/A007454/b007454.txt">Table of n, a(n) for n = 1..1000</a> (first 100 terms from Jean-François Alcover)
%H A007454 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 A007454 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 A007454 <a href="/index/Pos#posets">Index entries for sequences related to posets</a>
%F A007454 a(n) = A003430(n) - A007453(n) for n > 1. - _Sean A. Irvine_, Jan 04 2018
%t A007454 terms = 25; A[_] = 1;
%t A007454 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 A007454 A003430 = CoefficientList[A[x], x] // Rest;
%t A007454 mob[m_, n_] := If[Mod[m, n] == 0, MoebiusMu[m/n], 0];
%t A007454 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 A007454 Join[{1}, Rest[A003430 - EULERi[A003430]]] (* _Jean-François Alcover_, Jan 23 2020 *)
%Y A007454 Cf. A003430, A007453.
%K A007454 nonn
%O A007454 1,3
%A A007454 _N. J. A. Sloane_, _Mira Bernstein_
%E A007454 Name corrected by _Salah Uddin Mohammad_, Jun 07 2020