A002032 Number of n-colored connected graphs on n labeled nodes. (Formerly M2141 N0852)

1, 1, 2, 24, 912, 87360, 19226880, 9405930240, 10142439229440, 24057598104207360, 125180857812868300800, 1422700916050060841779200, 35136968950395142864227532800, 1876028272361273394915958613606400, 215474119792145796020405035320528076800

Offset

0,3

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

Andrew Howroyd, Table of n, a(n) for n = 0..50

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

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, May 29 2013

Name clarified by Andrew Howroyd, Dec 03 2018

a(0)=1 prepended by Andrew Howroyd, Jan 05 2024

Status

approved