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A339154
Number of essentially series oriented series-parallel networks with n elements and without unit elements in parallel.
3
0, 1, 1, 1, 3, 6, 14, 30, 70, 165, 397, 961, 2368, 5875, 14722, 37134, 94312, 240823, 618147, 1593606, 4125218, 10717064, 27934867, 73032798, 191464677, 503218042, 1325678981, 3499913710, 9258627528, 24538328431, 65147600774, 173243773337, 461400769439
OFFSET
1,5
COMMENTS
A series configuration is 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.
FORMULA
G.f.: P(x)^2/(1 - P(x)) where P(x) is the g.f. of A339155.
G.f.: B(x)^2/(1 + B(x)) where B(x) is the g.f. of A339156.
EXAMPLE
In the following examples, elements in series are juxtaposed and elements in parallel are separated by '|'. The unit element is denoted by 'o'.
a(2) = 1: (oo).
a(3) = 1: (ooo).
a(4) = 1: (oooo).
a(5) = 3: (ooooo), (o(oo|oo)), ((oo|oo)o).
a(6) = 6: (oooooo), (oo(oo|oo)), (o(oo|oo)o), ((oo|oo)oo), (o(oo|ooo)), ((oo|ooo)o).
PROG
(PARI) EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
seq(n)={my(p=O(x^2)); for(n=2, n, p=x+x*Ser(EulerT(Vec(p, 1-n))); p=p^2/(1+p)); Vec(p, -n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Nov 26 2020
STATUS
approved