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A108243
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a(n) = number of 3-regular (trivalent) multi-graphs without loops on 2n vertices; a(n) = number of symmetric 2n X 2n matrices with {0,1,2,3}-entries with row sum equal to 3 for each row and trace 0.
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6
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1, 1, 10, 760, 190050, 103050570, 102359800620, 168076482974400, 424343374430075100, 1560473478516337885500, 8014685021084051980870200, 55595731825871742484530751200, 506777617936508379069463525671000, 5933390819918520195635187162608235000, 87521940468361373047495526366554342050000
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OFFSET
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0,3
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LINKS
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FORMULA
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Linear differential equation satisfied by exponential generating function: {D(F)(0) = 1, (41580*t^5-3780*t^4+120*t^2+33*t-3)*F(t) + (498960*t^6-162540*t^5-11340*t^4+3+1350*t^3-60*t+132*t^2)*(d/dt)F(t) + (831600*t^7-466200*t^6-30240*t^5+7410*t^4+44*t^3-81*t^2)*(d^2/dt^2)F(t) + (443520*t^8-352800*t^7-18144*t^6+7372*t^5-18*t^3)*(d^3/dt^3)F(t) + (95040*t^9-97920*t^8-3456*t^7+1992*t^6)*(d^4/dt^4)F(t) + (8448*t^10-10688*t^9-192*t^8+144*t^7)*(d^5/dt^5)F(t) + (256*t^11-384*t^10)*(d^6/dt^6)F(t),
with F(0) = 1, `@@`(D, 5)(F)(0) = 103050570, `@@`(D, 2)(F)(0) = 10, `@@`(D, 3)(F)(0) = 760, `@@`(D, 4)(F)(0) = 190050}
Linear recurrence satisfied by a(n): {(4989600 + 5718768*n^7 + 1045440*n^8 + 123200*n^9 + 8448*n^10 + 256*n^11 + 30135960*n + 75458988*n^2 + 105258076*n^3 + 91991460*n^4 + 53358140*n^5 + 21100464*n^6)*a(n) + (-19958400 - 1534368*n^7 - 182592*n^8 - 12608*n^9 - 384*n^10 - 75637440*n - 125414712*n^2 - 119890252*n^3 - 73239888*n^4 - 29906772*n^5 - 8276184*n^6)*a(n + 1) + (-4989600 - 5760*n^7 - 192*n^8 - 11840760*n - 12084468*n^2 - 6932520*n^3 - 2446668*n^4 - 544320*n^5 - 74592*n^6)*a(n + 2) + (1857240 + 144*n^7 + 3447358*n + 2724762*n^2 + 1186966*n^3 + 307470*n^4 + 47332*n^5 + 4008*n^6)*a(n + 3) + (5445 + 3289*n + 660*n^2 + 44*n^3)*a(n + 4) + (-3003 - 1635*n - 297*n^2 - 18*n^3)*a(n + 5) + 3*a(n + 6),
with a(0) = 1, a(1) = 1, a(2) = 10, a(3) = 760, a(4) = 190050, a(5) = 103050570}
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EXAMPLE
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a(1)=1 is the graph on 1, 2 with three copies of the edge (1,2).
a(2)=10 are relabelings of the graphs on 1,2,3,4:
K_4 x 1
+ {(1,2), (1,2), (1,3), (3,4), (3,4), (2,4)} x 6 relabelings
+ {(1,2), (1,2), (1,2), (3,4), (3,4), (3,4)} x 3 relabelings.
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CROSSREFS
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Even bisection of column k=3 of A333351.
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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