

A002032


Number of ncolored 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

Table of n, a(n) for n=1..14.
R. C. Read, E. M. Wright, Colored graphs: A correction and extension, Canad. J. Math. 22 1970 594596.


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*(nr))*M_r(k1) [M_n(k) = A322280(n,k)], m_n(k) = M_n(k) Sum_{r=1..n1} C(n1,r1)*m_r(k)*M_{nr}(k) [m_n(k) = A322279(n,k)], f_n(k) = Sum_{r=1..k} (1)^(kr)*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 A002027A002030, A002031.  Andrew Howroyd, Dec 03 2018


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


STATUS

approved



