cp's OEIS Frontend

A339159 Number of achiral series-parallel networks with n elements.

%I A339159 #8 Nov 28 2020 19:55:09
%S A339159 1,2,3,7,12,29,54,130,258,616,1274,3030,6458,15287,33335,78694,174587,
%T A339159 411469,925246,2179010,4952389,11662221,26733827,62980863,145385388,
%U A339159 342766624,795810810,1878109984,4381423357,10352044123,24247955489,57362089607
%N A339159 Number of achiral series-parallel networks with n elements.
%C A339159 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 series or parallel configurations with n unit elements that are invariant under the reversal of all contained series configurations.
%F A339159 a(n) = A339157(n) + A339158(n) for n > 1.
%e A339159 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'.
%e A339159 a(1) = 1: (o).
%e A339159 a(2) = 2: (oo), (o|o).
%e A339159 a(3) = 3: (ooo), (o|oo), (o|o|o), (o|ooo), (oo|oo), (o|o|oo), (o|o|o|o).
%e A339159 a(4) = 7: (oooo), ((o|o)(o|o)), (o(o|o)o).
%o A339159 (PARI) \\ here B(n) gives A003430 as a power series.
%o A339159 EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
%o A339159 B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}
%o A339159 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+O(x*x^n))}
%Y A339159 Cf. A003430 (oriented), A339157, A339158, A339225 (unoriented).
%K A339159 nonn
%O A339159 1,2
%A A339159 _Andrew Howroyd_, Nov 27 2020