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A046860 Triangle giving a(n,k) = number of k-colored labeled graphs with n nodes. 5
1, 1, 4, 1, 24, 48, 1, 160, 1152, 1536, 1, 1440, 30720, 122880, 122880, 1, 18304, 1152000, 10813440, 29491200, 23592960, 1, 330624, 65630208, 1348730880, 7707033600, 15854469120, 10569646080, 1, 8488960, 5858721792, 261070258176, 2853804441600, 11499774935040, 18940805775360, 10823317585920 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

LINKS

Alois P. Heinz, Rows n = 1..50, flattened

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

FORMULA

a(n, k) = Sum_{r=1..n-1} C(n, r) 2^(r*(n-r)) a(r, k-1).

1 + Sum_{n>=1} Sum_{k=1..n} a(n,k)*y^k*x^n/(n!*2^C(n,2)) = 1/(1-y(E(x)-1)) where E(x) = Sum_{n>=0} x^n/(n!*2^C(n,2)). - Geoffrey Critzer, May 06 2020

EXAMPLE

Triangle begins:

  1;

  1,     4;

  1,    24,      48;

  1,   160,    1152,     1536;

  1,  1440,   30720,   122880,   122880;

  1, 18304, 1152000, 10813440, 29491200, 23592960;

  ...

MAPLE

a:= proc(n, k) option remember; `if`([n, k]=[0$2], 1,

      add(binomial(n, r)*2^(r*(n-r))*a(r, k-1), r=0..n-1))

    end:

seq(seq(a(n, k), k=1..n), n=1..8);  # Alois P. Heinz, Apr 21 2020

MATHEMATICA

a[n_ /; n >= 1, k_ /; k >= 1] := a[n, k] = Sum[ Binomial[n, r]*2^(r*(n - r))*a[r, k - 1], {r, 1, n - 1}]; a[_, 0] = 1; Flatten[ Table[ a[n, k], {n, 1, 8}, {k, 0, n - 1}]] (* Jean-François Alcover, Dec 12 2011, after formula *)

CROSSREFS

Column #1 gives A000683.

Main diagonal gives A011266.

Row sums give A334282.

Cf. A000683, A006201, A006202.

Sequence in context: A079621 A285061 A285066 * A089505 A300083 A062328

Adjacent sequences:  A046857 A046858 A046859 * A046861 A046862 A046863

KEYWORD

tabl,easy,nice,nonn

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Vladeta Jovovic, Feb 04 2000

STATUS

approved

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Last modified August 1 00:13 EDT 2021. Contains 346377 sequences. (Running on oeis4.)