A339290 - OEIS

%I #11 Dec 22 2020 17:53:24

%S 1,1,2,5,13,36,103,306,930,2887,9100,29082,93951,306414,1007361,

%T 3335088,11108986,37203873,125193694,423099557,1435427202,4886975378,

%U 16690971648,57172387872,196358421066,676050576441,2332887221847,8067160995797,27950871439353,97019613539949

%N Number of oriented series-parallel networks with n elements and without multiple unit elements in parallel.

%C A series configuration is an ordered concatenation of two or more parallel configurations and a parallel configuration is a multiset of two or more unit elements or series configurations. In this variation, parallel configurations may include the unit element only once. a(n) is the total number of series and parallel configurations with n unit elements.

%H Andrew Howroyd, <a href="/A339290/b339290.txt">Table of n, a(n) for n = 1..500</a>

%F a(n) = A339288(n) + A339289(n).

%F G.f.: P(x)/(1 - P(x)) where P(x) is the g.f. of A339289.

%e In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.

%e a(1) = 1: (o).

%e a(2) = 1: (oo).

%e a(3) = 2: (ooo), (o|oo).

%e a(4) = 5: (oooo), (o(o|oo)), ((o|oo)o), (o|ooo), (oo|oo).

%e a(5) = 13: (ooooo), (oo(o|oo)), (o(o|oo)o), ((o|oo)oo), (o(o|ooo)), (o(oo|oo)), ((o|ooo)o), ((oo|oo)o), (o|oooo), (o|o(o|oo)), (o|(o|oo)o), (oo|ooo), (o|oo|oo).

%o (PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}

%o seq(n, Z=x)={my(p=Z+O(x^2)); for(n=2, n, p = Z + (1 + Z)*x*Ser(EulerT( Vec(p^2/(1+p), -n) ))); Vec(p)}

%Y A003430 is the case with multiple unit elements in parallel allowed.

%Y A058387 is the case that order is not significant in series configurations.

%Y Cf. A339156, A339288, A339289, A339293 (achiral), A339296 (unoriented), A339301 (labeled).

%K nonn

%O 1,3

%A _Andrew Howroyd_, Dec 07 2020