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A339296
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Number of unoriented 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, 4, 9, 23, 60, 170, 496, 1505, 4665, 14772, 47415, 154093, 505394, 1671009, 5561274, 18615687, 62624016, 211605003, 717823582, 2443711716, 8345934897, 28587110560, 98181058020, 338029066457, 1166451261583, 4033596172701, 13975467586797, 48509872173875
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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 distinct series or parallel configurations with n unit elements modulo reversing the order of all 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) = 4: (oooo), (o(o|oo)), (o|ooo), (oo|oo).
a(5) = 9: (ooooo), (oo(o|oo)), (o(o|oo)o), (o(o|ooo)), (o(oo|oo)), (o|oooo), (o|o(o|oo)), (oo|ooo), (o|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 A339225.
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+B(n, Z))/2}
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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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