A339293 Number of achiral series-parallel networks with n elements and without multiple unit elements in parallel.

1, 1, 2, 3, 5, 10, 17, 34, 62, 123, 230, 462, 879, 1772, 3427, 6930, 13562, 27501, 54338, 110449, 219962, 448054, 898146, 1833248, 3694974, 7556473, 15301319, 31349605, 63734241, 130807801, 266853663, 548599872, 1122544408, 2311386319, 4742103354, 9778950947

Offset: 1

Author

Andrew Howroyd, Dec 07 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. In this variation, parallel configurations may include the unit element only once. a(n) is the number of series or parallel configurations with n unit elements that are invariant under the reversal of all contained series configurations.

Examples

			In the following examples, 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) = 2: (ooo), (o|oo).
a(4) = 3: (oooo), (o|ooo), (oo|oo).
a(5) = 5: (ooooo), (o(o|oo)o), (o|oooo), (oo|ooo), (o|oo|oo).
a(6) = 10: (oooooo), ((o|oo)(o|oo)), (o(o|ooo)o), (o(oo|oo)o), (o|ooooo), (o|o(o|oo)o), (oo|oooo), (ooo|ooo), (o|oo|ooo), (oo|oo|oo).
		

Crossrefs

Cf. A339159, A339290 (oriented), A339291, A339292, A339296 (unoriented).

Programs

PARI

\\ here B(n) gives A339290 as a power series.
\\ Note replacing Z by x/(1-x) gives A339159.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(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) ))); p}
seq(n, Z=x)={my(q=subst(B((n+1)\2, Z), x, x^2), s=q^2/(1+q), p=Z+O(x^2), t=0); forstep(n=2, n, 2, t=q*(1 + p); p=Z + (1 + Z)*x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2, -n-1))) - t); Vec(p+t+O(x*x^n))}

Formula

a(n) = A339291(n) + A339292(n) for n > 1.