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The number of connected, unlabeled, undirected, edge-signed cubic graphs (admitting loops and multiedges) on 2n vertices where the degree of the first sign is 2 at each node.
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%I #12 May 05 2023 12:54:18

%S 1,2,5,19,88,553,4619,49137,646815,10053183,178725865,3555840644,

%T 78048875298,1871066903575,48617053973267,1360733669185473,

%U 40810827325698897,1305690378666580997,44387116312631271929,1597768080980647428027,60710507893875818581964

%N The number of connected, unlabeled, undirected, edge-signed cubic graphs (admitting loops and multiedges) on 2n vertices where the degree of the first sign is 2 at each node.

%C Obtained from the cubic graphs A005967 (connected undirected cubic graphs that may have loops and/or multiedges) by signing each edge with a plus or a minus such that two pluses and one minus meet at each vertex.

%H Richard J. Mathar, <a href="http://vixra.org/abs/1901.0148">Feynman diagrams of the QED vacuum polarization</a>, vixra:1901.0148 (2019) Section II.

%o (PARI) \\ See A339645 for combinatorial species functions.

%o cycleIndexSeries(n)={1+sLog(sCartProd(sExp(dihedralGroupSeries(n)), sExp(symGroupCycleIndex(2)*x^2 + O(x*x^n))))}

%o seq(n)={Vec(substpol(OgfSeries(cycleIndexSeries(2*n)), x^2, x))} \\ _Andrew Howroyd_, May 05 2023

%Y Cf. A005967 (unsigned), A054499 (only one cycle of pluses), A170946 (directed plus-edges).

%K nonn

%O 0,2

%A _R. J. Mathar_, Jan 13 2019

%E Terms a(6) and beyond from _Andrew Howroyd_, May 05 2023