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A339157
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Number of essentially series achiral series-parallel networks with n elements.
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5
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1, 1, 1, 3, 4, 11, 17, 46, 78, 203, 372, 946, 1830, 4561, 9207, 22609, 47166, 114514, 245154, 590345, 1289950, 3087959, 6858746, 16352074, 36800928, 87502317, 199036637, 472483088, 1084108363, 2571356964, 5942191918, 14090541799, 32754720101, 77684033014, 181473276607
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OFFSET
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1,4
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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. a(n) is the number of series 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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G.f.: x + (1 + P(x))*B(x^2) where P(x) is the g.f. of A339158 and B(x) is the g.f. of A003430.
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EXAMPLE
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In the following examples of series-parallel networks, 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) = 1: (ooo).
a(4) = 3: (oooo), ((o|o)(o|o)), (o(o|o)o).
a(5) = 4: (ooooo), ((o|o)o(o|o)), (o(o|oo)o), (o(o|o|o)o).
a(6) = 11: (oooooo), ((o|o)oo(o|o)), (o(o|o)(o|o)o), ((o|oo)(o|oo)), ((o|o|o)(o|o|o)), (oo(o|o)oo), ((o|o)(o|o)(o|o)), (o(o|ooo)o), (o(oo|oo)o), (o(o|o|oo)o), (o(o|o|o|o)o).
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PROG
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(PARI) \\ here B(n) gives A003430 as a power series.
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
B(n)={my(p=x+O(x^2)); for(n=2, n, p=x*Ser(EulerT(Vec(p^2/(1+p)+x)))); p}
seq(n)={my(q=subst(B((n+1)\2), x, x^2), s=x^2+q^2/(1+q), p=x+O(x^2)); for(n=1, n\2, p = x + q*(1 + x + x*Ser(EulerT(Vec(p+(s-subst(p, x, x^2))/2))) - p)); Vec(p+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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