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A349276
Number of unlabeled P-series with n elements.
4
1, 2, 5, 13, 31, 76, 178, 423, 988, 2312, 5361, 12427, 28626, 65813, 150700, 344232, 783832, 1780650, 4034591, 9121571, 20576349, 46322816, 104079338, 233421517, 522574991, 1167974002, 2606282841, 5806953923, 12919314397, 28702716868, 63682839588, 141111193270
OFFSET
1,2
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.
A poset is called a P-graph if it can be expressed as the ordinal sum of the antichain posets (including the singleton poset).
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.
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'.
FORMULA
a(n) = A255047(n-1) + A349488(n).
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
MAPLE
max(1, 2^(d-1)-1), d=numtheory[divisors](j)), j=1..n)/n)
end:
seq(a(n), n=1..30); # Alois P. Heinz, Jan 05 2022
MATHEMATICA
a[n_] := a[n] = If[n == 0, 1, Sum[a[n - j]*Sum[d*
Max[1, 2^(d - 1) - 1], {d, Divisors[j]}], {j, 1, n}]/n];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Mar 18 2022, after Alois P. Heinz *)
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={EulerT(Vec((1 -2*x +2*x^2)/((1-x)*(1-2*x)) + O(x*x^n)))} \\ Andrew Howroyd, Nov 19 2021
CROSSREFS
Cf. A003430 (series-parallel posets), A255047, A349488.
Sequence in context: A063636 A076501 A369434 * A307569 A200772 A099515
KEYWORD
nonn
AUTHOR
STATUS
approved

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Last modified September 20 20:48 EDT 2024. Contains 376078 sequences. (Running on oeis4.)