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%I #5 Dec 23 2020 13:54:06
%S 1,4,15,105,873,9997,134582,2096206,36391653,693779666,14346005530,
%T 319042302578,7579064231400,191264021808301,5103735168371201,
%U 143438421861618397,4231407420255210941,130633362289335958866,4209546674788934624394,141259712052820378949746
%N Number of inequivalent colorings of unoriented series-parallel networks with n colored elements.
%C Equivalence is up to permutation of the colors and reversal of all parts combined in series. Any number of colors may be used. See A339282 for additional details.
%e In the following examples elements in series are juxtaposed and elements in parallel are separated by '|'.
%e a(1) = 1: (1).
%e a(2) = 4: (11), (12), (1|1), (1|2).
%e a(3) = 15: (111), (112), (121), (123), (1(1|1)), (1(1|2)), (1(2|2)), (1(2|3)), (1|1|1), (1|1|2), (1|2|3), (1|11), (1|12), (1|22), (1|23).
%o (PARI) \\ See links in A339645 for combinatorial species functions.
%o B(n)={my(Z=x*sv(1), p=Z+O(x^2)); for(n=2, n, p=sEulerT(p^2/(1+p) + Z)-1); p}
%o cycleIndexSeries(n)={my(Z=x*sv(1), q=sRaise(B((n+1)\2), 2), s=x^2*sv(2)+q^2/(1+q), p=Z+O(x^2), t=p); for(n=1, n\2, t=Z + q*(1 + p); p=Z + sEulerT(t+(s-sRaise(t, 2))/2) - t - 1); (p+t-Z+B(n))/2}
%o InequivalentColoringsSeq(cycleIndexSeries(15))
%Y Cf. A339225 (uncolored), A339233 (oriented), A339280, A339282, A339283, A339645.
%K nonn
%O 1,2
%A _Andrew Howroyd_, Dec 22 2020
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