A006201 Number of colorings of labeled graphs on n nodes using exactly 3 colors, divided by 48. (Formerly M5167)

0, 0, 1, 24, 640, 24000, 1367296, 122056704, 17282252800, 3897054412800, 1400795928395776, 802530102499344384, 732523556206878392320, 1064849635418836398243840, 2464403435614136308036796416

Offset

1,4

Comments

Equals 1/48*A213442. - Peter Bala, Apr 12 2013

References

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 18, table 1.5.1, column 3 (divided by 8).

R. C. Read, personal communication.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Alois P. Heinz, Table of n, a(n) for n = 1..75

R. C. Read, The number of k-colored graphs on labelled nodes, Canad. J. Math., 12 (1960), 410—414.

R. C. Read, Letter to N. J. A. Sloane, Oct. 29, 1976

Formula

Let E(x) = sum {n >= 0} x^n/(n!*2^C(n,2)) = 1 + x + x^2/(2!*2) + x^3/(3!*2^3) + x^4/(4!*2^6) + .... Then a generating function is 1/48*(E(x) - 1)^3 = x^3/(3!*2^3) + 24*x^4/(4!*2^6) + 640*x^6/(5!*2^10) + ... (see Read). - Peter Bala, Apr 12 2013

Mathematica

F2[n_] := Sum[Binomial[n, r]*2^(r*(n-r)), {r, 1, n-1}]; F3[n_] := Sum[Binomial[n, r]*2^(r*(n-r))*F2[r], {r, 1, n-1}]; a[n_] := F3[n]/48; Table[a[n], {n, 1, 15}] (* Jean-François Alcover, Mar 06 2014, after Maple code in A213442 *)

Prog

(PARI) seq(n)={Vec(serconvol(sum(j=1, n, x^j*j!*2^binomial(j, 2)) + O(x*x^n), (sum(j=1, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n))^3)/48, -n)} \\ Andrew Howroyd, Nov 30 2018

Crossrefs

Cf. A000683. A diagonal of A058843. A213442.

Sequence in context: A331322 A126153 A002553 * A118051 A208441 A231449

Adjacent sequences: A006198 A006199 A006200 * A006202 A006203 A006204

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Vladeta Jovovic, Feb 03 2000

Status

approved