A339157 Number of essentially series achiral series-parallel networks with n elements.

1, 1, 1, 3, 4, 11, 17, 46, 78, 203, 372, 946, 1830, 4561, 9207, 22609, 47166, 114514, 245154, 590345, 1289950, 3087959, 6858746, 16352074, 36800928, 87502317, 199036637, 472483088, 1084108363, 2571356964, 5942191918, 14090541799, 32754720101, 77684033014, 181473276607

Offset: 1

Author

Andrew Howroyd, Nov 27 2020

Keywords

• nonn

Comments

A series configuration is the unit element or 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 series configurations with n unit elements that are invariant under the reversal of all contained series configurations.

Examples

			In the following examples of series-parallel networks, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (oo).
a(3) = 1: (ooo).
a(4) = 3: (oooo), ((o|o)(o|o)), (o(o|o)o).
a(5) = 4: (ooooo), ((o|o)o(o|o)), (o(o|oo)o), (o(o|o|o)o).
a(6) = 11: (oooooo), ((o|o)oo(o|o)), (o(o|o)(o|o)o), ((o|oo)(o|oo)), ((o|o|o)(o|o|o)), (oo(o|o)oo), ((o|o)(o|o)(o|o)), (o(o|ooo)o), (o(oo|oo)o), (o(o|o|oo)o), (o(o|o|o|o)o).
		

Crossrefs

Cf. A003430, A007453 (oriented), A339158, A339159, A339223 (unoriented).

Programs

PARI

\\ here B(n) gives A003430 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}
seq(n)={my(q=subst(B((n+1)\2), x, x^2), s=x^2+q^2/(1+q), p=x+O(x^2)); for(n=1, n\2, p = x + q*(1 + x + x*Ser(EulerT(Vec(p+(s-subst(p,x,x^2))/2))) - p)); Vec(p+O(x*x^n))}

Formula

G.f.: x + (1 + P(x))*B(x^2) where P(x) is the g.f. of A339158 and B(x) is the g.f. of A003430.