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%I #37 Mar 18 2022 04:33:07
%S 1,2,5,13,31,76,178,423,988,2312,5361,12427,28626,65813,150700,344232,
%T 783832,1780650,4034591,9121571,20576349,46322816,104079338,233421517,
%U 522574991,1167974002,2606282841,5806953923,12919314397,28702716868,63682839588,141111193270
%N Number of unlabeled P-series with n elements.
%C The class of all P-series is a subclass of the class of series-parallel posets and it contains the class of P-graphs as a subclass.
%C A poset is called a P-graph if it can be expressed as the ordinal sum of the antichain posets (including the singleton poset).
%C A poset is called a P-series if it is either a P-graph or it can be expressed as the direct sum of the P-graphs.
%C For example, all the 3-element posets are P-series, where only the connected posets and the antichains are P-graphs. On the other hand, the 4-element poset <{x,y,z,w},{x<.z, z<.w, y<.w, x||y, y||z}> and its dual are both series-parallel which are not the P-series. Here, by 'x<.z' we mean 'x is covered by z'.
%H Alois P. Heinz, <a href="/A349276/b349276.txt">Table of n, a(n) for n = 1..3217</a>
%F a(n) = A255047(n-1) + A349488(n).
%F G.f: -1 + exp(Sum_{k>=1} B(x^k)/k) where B(x) = x*(1 - 2*x + 2*x^2)/((1 - x)*(1 - 2*x)). - _Andrew Howroyd_, Jan 06 2022
%p a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)*add(d*
%p max(1, 2^(d-1)-1), d=numtheory[divisors](j)), j=1..n)/n)
%p end:
%p seq(a(n), n=1..30); # _Alois P. Heinz_, Jan 05 2022
%t a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j]*Sum[d*
%t Max[1, 2^(d - 1) - 1], {d, Divisors[j]}], {j, 1, n}]/n];
%t Table[a[n], {n, 1, 30}] (* _Jean-François Alcover_, Mar 18 2022, after _Alois P. Heinz_ *)
%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o seq(n)={EulerT(Vec((1 -2*x +2*x^2)/((1-x)*(1-2*x)) + O(x*x^n)))} \\ _Andrew Howroyd_, Nov 19 2021
%Y Cf. A003430 (series-parallel posets), A255047, A349488.
%K nonn
%O 1,2
%A _Salah Uddin Mohammad_, Nov 12 2021
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