A339155 - OEIS

%I #8 Nov 28 2020 01:21:46

%S 1,0,0,1,1,3,5,13,29,70,165,409,1001,2505,6278,15904,40447,103567,

%T 266229,687668,1782573,4637731,12103112,31679212,83135973,218713492,

%U 576683119,1523740365,4033915677,10698680606,28422818782,75629586540,201539697208,537818080714

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

%C A series configuration is an ordered concatenation of two or more parallel configurations and a parallel configuration is the unit element or a multiset of two or more series configurations. a(n) is the number of parallel configurations with n unit elements.

%F G.f.: B(x)/(1 + B(x)) where B(x) is the g.f. of A339156.

%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(4) = 1: (oo|oo).

%e a(5) = 1: (oo|ooo).

%e a(6) = 3: (oo|oooo), (ooo|ooo), (oo|oo|oo).

%e a(7) = 4: (oo|ooooo), (oo|o(oo|oo)), (oo|(oo|oo)o), (ooo|oooo), (oo|oo|ooo).

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

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

%Y Cf. A003430, A339152, A339154, A339156.

%K nonn

%O 1,6

%A _Andrew Howroyd_, Nov 26 2020