A243394
Number of isomorphism classes of connected 3-regular multigraphs with n vertices and with loops and semi-edges allowed.
Original entry on oeis.org
2, 5, 7, 22, 43, 141, 373, 1270, 4053, 14671, 52826, 203289, 795581, 3241367, 13504130, 57904671, 253856990, 1139231977, 5219113084, 24401837085, 116278408069, 564380686932, 2787884851040, 14007277302822, 71538337097031, 371197207327709, 1955833646495247, 10459788214042492
Offset: 1
A361412
Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an unoriented edge (or loop), loops allowed.
Original entry on oeis.org
1, 3, 12, 67, 441, 3464, 31616, 331997, 3961462, 53105424, 791237787, 12978022526, 232407307054, 4511887729886, 94385418177277, 2116529900006321, 50646269987874834, 1288091152941695791, 34697173459041347465, 986800102740080746702, 29548269236430810895013
Offset: 0
The a(1) = 3 multigraphs correspond to either a triple edge rooted on one of the edges or to a single edge with a loop at each end and rooted on either the edge or the loop.
A361446
Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop), loops allowed.
Original entry on oeis.org
1, 3, 16, 99, 717, 5964, 56701, 611750, 7432491, 100838222, 1514749135, 24989362186, 449429188211, 8754181791029, 183621843677724, 4126714250580949, 98932328702693666, 2520187379996442269, 67980528958530199837, 1935753445850303203221, 58025998739501873764826
Offset: 0
A361448
Number of connected 3-regular multigraphs on 2n unlabeled nodes rooted at an oriented edge (or loop) whose removal does not disconnect the graph, loops allowed.
Original entry on oeis.org
1, 2, 10, 66, 511, 4536, 45519, 512661, 6436571, 89505875, 1369509795, 22908806774, 416408493351, 8178599551905, 172690849144538, 3902128758180500, 93970611848528998, 2402929936231885063, 65029668312580777779, 1856984518220396165657, 55803367549204703645086
Offset: 0
a(2) = 10 = A361447(2) + 1 because there is one graph where the orientation of the rooted edge makes a difference:
1 __
/| \ | |
|| 3---4 |
\| / |__|
2
The nodes are labeled 1,2,3,4. There is a double edge between nodes 1 and 2 and a loop at node 4. Roots at the edges (1,3) and (3,1) are considered different because orientation is considered. Roots at (1,3) and (2,3) are considered the same because the resulting graphs are isomorphic. Roots at (3,4) or (4,3) are disallowed because the removal of that edge would disconnect the graph.
A366039
Irregular triangle read by rows: T(n,k) = number of cells of dimension k in the moduli space of tropical curves of genus n.
Original entry on oeis.org
1, 2, 2, 2, 1, 2, 5, 9, 12, 8, 5, 1, 3, 7, 21, 43, 75, 89, 81, 42, 17, 1, 3, 11, 34, 100, 239, 492, 784, 1002, 926, 632, 260, 71
Offset: 2
The irregular triangle T(n,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 11 12...
2: 1, 2, 2, 2;
3: 1, 2, 5, 9, 12, 8, 5;
4: 1, 3, 7, 21, 43, 75, 89, 81, 42, 17;
5: 1, 3, 11, 34, 100, 239, 492, 784, 1002, 926, 632, 260, 71;
...
A323389
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.
Original entry on oeis.org
1, 2, 5, 19, 88, 553, 4619, 49137, 646815, 10053183, 178725865, 3555840644, 78048875298, 1871066903575, 48617053973267, 1360733669185473, 40810827325698897, 1305690378666580997, 44387116312631271929, 1597768080980647428027, 60710507893875818581964
Offset: 0
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\\ See A339645 for combinatorial species functions.
cycleIndexSeries(n)={1+sLog(sCartProd(sExp(dihedralGroupSeries(n)), sExp(symGroupCycleIndex(2)*x^2 + O(x*x^n))))}
seq(n)={Vec(substpol(OgfSeries(cycleIndexSeries(2*n)), x^2, x))} \\ Andrew Howroyd, May 05 2023
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