A046860 Triangle giving a(n,k) = number of k-colored labeled graphs with n nodes.

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

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