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Number of essentially parallel unoriented series-parallel networks with n elements and without multiple unit elements in parallel.
4

%I #6 Dec 08 2020 02:27:37

%S 1,0,1,2,4,10,25,69,197,589,1806,5685,18168,58905,192904,637294,

%T 2119994,7094961,23865782,80642017,273571625,931389949,3181184007,

%U 10897272983,37429033777,128874546753,444744161951,1538030244174,5329246656885,18499283612755

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

%C See A339296 for additional details.

%F a(n) = (A339289(n) + A339292(n)) / 2.

%e In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.

%e a(1) = 1: (o).

%e a(3) = 1: (o|oo).

%e a(4) = 2: (o|ooo), (oo|oo).

%e a(5) = 4: (o|oooo), (o|o(o|oo)), (oo|ooo), (o|oo|oo).

%o (PARI) \\ here B(n) gives A339290 as a power series.

%o EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}

%o 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 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+1-1/(1+B(n,Z)))/2}

%Y Cf. A339224, A339289 (oriented), A339292 (achiral), A339294, A339296.

%K nonn

%O 1,4

%A _Andrew Howroyd_, Dec 07 2020