cp's OEIS Frontend

A339293 Number of achiral series-parallel networks with n elements and without multiple unit elements in parallel.

%I A339293 #8 Dec 08 2020 02:27:28
%S A339293 1,1,2,3,5,10,17,34,62,123,230,462,879,1772,3427,6930,13562,27501,
%T A339293 54338,110449,219962,448054,898146,1833248,3694974,7556473,15301319,
%U A339293 31349605,63734241,130807801,266853663,548599872,1122544408,2311386319,4742103354,9778950947
%N A339293 Number of achiral series-parallel networks with n elements and without multiple unit elements in parallel.
%C A339293 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. In this variation, parallel configurations may include the unit element only once. 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 A339293 a(n) = A339291(n) + A339292(n) for n > 1.
%e A339293 In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
%e A339293 a(1) = 1: (o).
%e A339293 a(2) = 1: (oo).
%e A339293 a(3) = 2: (ooo), (o|oo).
%e A339293 a(4) = 3: (oooo), (o|ooo), (oo|oo).
%e A339293 a(5) = 5: (ooooo), (o(o|oo)o), (o|oooo), (oo|ooo), (o|oo|oo).
%e A339293 a(6) = 10: (oooooo), ((o|oo)(o|oo)), (o(o|ooo)o), (o(oo|oo)o), (o|ooooo), (o|o(o|oo)o), (oo|oooo), (ooo|ooo), (o|oo|ooo), (oo|oo|oo).
%o A339293 (PARI) \\ here B(n) gives A339290 as a power series.
%o A339293 \\ Note replacing Z by x/(1-x) gives A339159.
%o A339293 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A339293 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}
%o A339293 seq(n, Z=x)={my(q=subst(B((n+1)\2, Z), x, x^2), s=q^2/(1+q), p=Z+O(x^2), t=0); forstep(n=2, n, 2, t=q*(1 + p); p=Z + (1 + Z)*x*Ser(EulerT(Vec(t+(s-subst(t, x, x^2))/2, -n-1))) - t); Vec(p+t+O(x*x^n))}
%Y A339293 Cf. A339159, A339290 (oriented), A339291, A339292, A339296 (unoriented).
%K A339293 nonn
%O A339293 1,3
%A A339293 _Andrew Howroyd_, Dec 07 2020