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A339293
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Number of achiral series-parallel networks with n elements and without multiple unit elements in parallel.
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4
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1, 1, 2, 3, 5, 10, 17, 34, 62, 123, 230, 462, 879, 1772, 3427, 6930, 13562, 27501, 54338, 110449, 219962, 448054, 898146, 1833248, 3694974, 7556473, 15301319, 31349605, 63734241, 130807801, 266853663, 548599872, 1122544408, 2311386319, 4742103354, 9778950947
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OFFSET
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1,3
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COMMENTS
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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.
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LINKS
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FORMULA
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EXAMPLE
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In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(1) = 1: (o).
a(2) = 1: (oo).
a(3) = 2: (ooo), (o|oo).
a(4) = 3: (oooo), (o|ooo), (oo|oo).
a(5) = 5: (ooooo), (o(o|oo)o), (o|oooo), (oo|ooo), (o|oo|oo).
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).
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PROG
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(PARI) \\ here B(n) gives A339290 as a power series.
\\ Note replacing Z by x/(1-x) gives A339159.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
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}
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))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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