A339291 Number of essentially series achiral series-parallel networks with n elements and without multiple unit elements in parallel.

0, 1, 1, 1, 2, 4, 6, 13, 21, 44, 76, 158, 281, 584, 1067, 2211, 4131, 8535, 16231, 33481, 64594, 133067, 259821, 534869, 1054751, 2170736, 4316320, 8884035, 17788985, 36627593, 73776883, 151996070, 307705669, 634411061, 1289890551, 2661708319

Offset: 1

Author

Andrew Howroyd, Dec 07 2020

Keywords

• nonn

Comments

See A339293 for additional details.

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(2) = 1: (oo).
a(3) = 1: (ooo).
a(4) = 1: (oooo).
a(5) = 2: (ooooo), (o(o|oo)o).
a(6) = 4: (oooooo), ((o|oo)(o|oo)), (o(o|ooo)o), (o(oo|oo)o).
		

Crossrefs

Cf. A339157, A339288 (oriented), A339290, A339292, A339293, A339294 (unoriented).

Programs

PARI

\\ here B(n) gives A339290 as a power series.
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=O(x^2)); forstep(n=2, n, 2, p=q*(1 + Z + (1 + Z)*x*Ser(EulerT(Vec(p+(s-subst(p, x, x^2))/2, 1-n))) - p)); Vec(p+O(x*x^n), -n)}

Formula

G.f.: (1 + P(x))*B(x^2) where P(x) is the g.f. of A339292 and B(x) is the g.f. of A339290.