cp's OEIS Frontend

A339291 Number of essentially series achiral series-parallel networks with n elements and without multiple unit elements in parallel.

%I A339291 #10 Dec 08 2020 15:23:34
%S A339291 0,1,1,1,2,4,6,13,21,44,76,158,281,584,1067,2211,4131,8535,16231,
%T A339291 33481,64594,133067,259821,534869,1054751,2170736,4316320,8884035,
%U A339291 17788985,36627593,73776883,151996070,307705669,634411061,1289890551,2661708319
%N A339291 Number of essentially series achiral series-parallel networks with n elements and without multiple unit elements in parallel.
%C A339291 See A339293 for additional details.
%F A339291 G.f.: (1 + P(x))*B(x^2) where P(x) is the g.f. of A339292 and B(x) is the g.f. of A339290.
%e A339291 In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
%e A339291 a(2) = 1: (oo).
%e A339291 a(3) = 1: (ooo).
%e A339291 a(4) = 1: (oooo).
%e A339291 a(5) = 2: (ooooo), (o(o|oo)o).
%e A339291 a(6) = 4: (oooooo), ((o|oo)(o|oo)), (o(o|ooo)o), (o(oo|oo)o).
%o A339291 (PARI) \\ here B(n) gives A339290 as a power series.
%o A339291 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A339291 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 A339291 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)); Vec(p+O(x*x^n), -n)}
%Y A339291 Cf. A339157, A339288 (oriented), A339290, A339292, A339293, A339294 (unoriented).
%K A339291 nonn
%O A339291 1,5
%A A339291 _Andrew Howroyd_, Dec 07 2020