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A239910
Number of forests with three connected components in the complete graph K_{n}.
5
0, 0, 1, 6, 45, 435, 5250, 76608, 1316574, 26100000, 587030895, 14780620800, 412069511139, 12604714327296, 419801484375000, 15123782440058880, 586049426860524300, 24307340986526810112, 1074495780444130114509, 50429952000000000000000
OFFSET
1,4
COMMENTS
Equation (47) of Liu-Chow (1984) also gives the analogous formulas for four and five components. (They should also be entered into the OEIS, in case someone wants to help.)
LINKS
C. J. Liu and Yutze Chow, On operator and formal sum methods for graph enumeration problems, SIAM J. Algebraic Discrete Methods, 5 (1984), no. 3, 384-406. MR0752043 (86d:05059).
FORMULA
From Harry Richman, Aug 17 2022: (Start)
a(n) = n^(n-6)*(n-1)*(n-2)*(n^2+13*n+60)/8.
E.g.f.: T(x)^{3}/3!, where T(x) is the e.g.f. for the number of spanning trees in K_{n} A000272, i.e., T(x) = Sum_{i>=1} i^(i-2)*x^i/i!. (End)
MAPLE
f := n-> (n-1)*(n-2)*n^(n-6)*(n^2+13*n+60)/8; [seq(f(n), n=3..20)];
MATHEMATICA
Table[(n-1)*(n-2) * n^(n - 6) * (n^2 + 13 n + 60)/8, {n, 1, 20}] (* Vincenzo Librandi, Apr 10 2014, simplified by Vaclav Kotesovec, Feb 20 2020 *)
PROG
(Magma) [(n-1)*(n-2)*n^(n-6)*(n^2+13*n+60)/8: n in [1..20]]; // Vincenzo Librandi, Apr 10 2014
CROSSREFS
Column m=3 of A105599. A diagonal of A138464. - Alois P. Heinz, Apr 10 2014
Sequence in context: A001879 A019577 A097814 * A374844 A228194 A331726
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Apr 09 2014
STATUS
approved