OFFSET
1,2
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 distinct series or parallel configurations with n unit elements modulo reversing the order of all series configurations.
FORMULA
EXAMPLE
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) = 2: (oo), (o|o).
a(3) = 4: (ooo), (o(o|o)), (o|o|o), (o|oo).
a(4) = 11: (oooo), (oo(o|o)), (o(o|o)o), ((o|o)(o|o)), (o(o|oo)), (o(o|o|o)), (o|o|o|o), (o|o|oo), (oo|oo), (o|ooo), (o|o(o|o)).
PROG
(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), t=p); for(n=1, n\2, t=x + q*(1 + p); p=x + x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2))) - t); Vec(p+t-x+B(n))/2}
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Nov 27 2020
STATUS
approved