A339153 Number of nonequivalent series-parallel networks with n elements and without unit elements in parallel.

1, 1, 1, 2, 3, 6, 10, 20, 37, 74, 144, 295, 594, 1229, 2540, 5324, 11177, 23684, 50326, 107593, 230743, 497008, 1073667, 2327213, 5057433, 11020414, 24068945, 52685541, 115555511, 253933732, 558993308, 1232569467, 2721958234, 6019784562, 13331192017, 29560633824

Offset: 1

Author

Andrew Howroyd, Nov 26 2020

Keywords

• nonn

Comments

Equivalence is up to rearrangement of the order of elements in both series and parallel configurations.

A series configuration is a multiset of two or more parallel configurations and a parallel configuration is a multiset of two or more series configurations. The unit element is considered to be a parallel configuration.

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

Crossrefs

Cf. A000669, A339151, A339152.

Programs

PARI

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
seq(n)={my(S=vector(n), P=vector(n)); P[1]=1; for(n=2, #S, my(t=EulerT(S[1..n])[n]); S[n]=EulerT(P[1..n])[n]; P[n]=t); S+P}

Formula

a(n) = A339151(n) + A339152(n).

Euler transform of A339152.

Euler transform of A339151 gives this sequence with a(1) = 0.