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 A002032 Number of n-colored connected graphs on n labeled nodes. (Formerly M2141 N0852) 5
 1, 2, 24, 912, 87360, 19226880, 9405930240, 10142439229440, 24057598104207360, 125180857812868300800, 1422700916050060841779200, 35136968950395142864227532800, 1876028272361273394915958613606400, 215474119792145796020405035320528076800 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Every connected graph on n nodes can be colored with n colors in exactly n! ways, so this sequence is just n! * A001187(n). - Andrew Howroyd, Dec 03 2018 REFERENCES N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence). N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence). LINKS R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594-596. FORMULA a(n) = n!*A001187(n). - Andrew Howroyd, Dec 03 2018 Define M_0(k)=1, M_n(0)=0, M_n(k) = Sum_{r=0..n} C(n,r)*2^(r*(n-r))*M_r(k-1) [M_n(k) = A322280(n,k)], m_n(k) = M_n(k) -Sum_{r=1..n-1} C(n-1,r-1)*m_r(k)*M_{n-r}(k) [m_n(k) = A322279(n,k)], f_n(k) = Sum_{r=1..k} (-1)^(k-r)*C(k,r)*m_n(r). This sequence gives a(n) = f_n(n). - Sean A. Irvine, May 29 2013, edited Andrew Howroyd, Dec 03 2018 The above formula is referenced by sequences A002027-A002030, A002031. - Andrew Howroyd, Dec 03 2018 MATHEMATICA (* b = A001187 *) b[n_] := b[n] = If[n == 0, 1, 2^(n(n-1)/2) - Sum[k* Binomial[n, k]*2^((n-k)(n-k-1)/2)*b[k], {k, 1, n-1}]/n]; a[n_] := n! b[n]; Array[a, 14] (* Jean-François Alcover, Aug 16 2019, using Alois P. Heinz's code for A001187 *) PROG (PARI) seq(n) = {Vec(serlaplace(serlaplace(1 + log(sum(k=0, n, 2^binomial(k, 2)*x^k/k!, O(x*x^n))))))} \\ Andrew Howroyd, Dec 03 2018 CROSSREFS Cf. A002027. A002028, A002029, A002030, A002031. Cf. A001187, A322278, A322279, A322280. Sequence in context: A012186 A012081 A137274 * A015212 A012228 A062029 Adjacent sequences: A002029 A002030 A002031 * A002033 A002034 A002035 KEYWORD nonn AUTHOR EXTENSIONS More terms from Sean A. Irvine, May 29 2013 Name clarified by Andrew Howroyd, Dec 03 2018 STATUS approved

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Last modified April 1 03:44 EDT 2023. Contains 361673 sequences. (Running on oeis4.)