A339223 Number of essentially series unoriented series-parallel networks with n elements.

1, 1, 2, 6, 17, 57, 196, 723, 2729, 10638, 42161, 169912, 692703, 2853523, 11852644, 49592966, 208800209, 883970867, 3760605627, 16068272965, 68925340187, 296705390322, 1281351319402, 5549911448062, 24103086681839, 104938476264310, 457920147387969, 2002462084788769

Offset: 1

Author

Andrew Howroyd, Nov 27 2020

Keywords

• nonn

Comments

See A339225 for additional details.

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

Crossrefs

Cf. A003430, A007453 (oriented), A339157 (achiral), A339224, A339225.

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+x+subst(x^2/(1+x),x,B(n)))/2}

Formula

a(n) = (A007453(n) + A339157(n))/2.