cp's OEIS Frontend

A339294 Number of essentially series unoriented series-parallel networks with n elements and without multiple unit elements in parallel.

%I A339294 #8 Dec 08 2020 15:23:42
%S A339294 0,1,1,2,5,13,35,101,299,916,2859,9087,29247,95188,312490,1033715,
%T A339294 3441280,11520726,38758234,130962986,444251957,1512321767,5164750890,
%U A339294 17689837577,60752024243,209154519704,721707099632,2495565928527,8646220929912,30010588561120
%N A339294 Number of essentially series unoriented series-parallel networks with n elements and without multiple unit elements in parallel.
%C A339294 See A339296 for additional details.
%F A339294 a(n) = (A339288(n) + A339291(n)) / 2.
%e A339294 In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
%e A339294 a(2) = 1: (oo).
%e A339294 a(3) = 1: (ooo).
%e A339294 a(4) = 2: (oooo), (o(o|oo)).
%e A339294 a(5) = 5: (ooooo), (oo(o|oo)), (o(o|oo)o), (o(o|ooo)), (o(oo|oo)).
%o A339294 (PARI) \\ here B(n) gives A339290 as a power series.
%o A339294 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A339294 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 A339294 seq(n, Z=x)={my(q=subst(B((n+1)\2, Z), x, x^2), s=q^2/(1+q), p=O(x^2)); forstep(n=2, n, 2, p=q*(1 + Z + (1 + Z)*x*Ser(EulerT(Vec(p+(s-subst(p, x, x^2))/2, 1-n))) - p)); my(t=B(n, Z)); Vec(p + t - t/(1+t), -n)/2}
%Y A339294 Cf. A339157, A339288 (oriented), A339290, A339291 (achiral), A339295, A339296.
%K A339294 nonn
%O A339294 1,4
%A A339294 _Andrew Howroyd_, Dec 07 2020