A219570 Triangular array read by rows. T(n,k) is the number of necklaces (turning over is not allowed) of n labeled black or white beads having exactly k black beads.

0, 1, 1, 1, 2, 1, 2, 6, 6, 2, 6, 24, 36, 24, 6, 24, 120, 240, 240, 120, 24, 120, 720, 1800, 2400, 1800, 720, 120, 720, 5040, 15120, 25200, 25200, 15120, 5040, 720, 5040, 40320, 141120, 282240, 352800, 282240, 141120, 40320, 5040, 40320, 362880, 1451520, 3386880, 5080320, 5080320, 3386880, 1451520, 362880, 40320

Offset: 0

Author

Geoffrey Critzer, Nov 23 2012

Keywords

• nonn
• tabl

Comments

Row sums are A066318.

Examples

			0;
1,      1;
1,      2,     1;
2,      6,     6,     2;
6,     24,    36,    24,     6;
24,   120,   240,   240,   120,    24;
120,  720,  1800,  2400,  1800,   720,  120;
720, 5040, 15120, 25200, 25200, 15120, 5040, 720;
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1274

Programs

Mathematica

nn=8;f[list_]:=Select[list,#>0&];Map[f,Drop[Range[0,nn]!CoefficientList[Series[Log[1/(1-(y+1)x)],{x,0,nn}],{x,y}],1]]//Grid

PARI

T(n, k) = if(n>0, (n-1)! * binomial(n, k)); \\ Andrew Howroyd, Oct 11 2017

Formula

E.g.f.: log(1/(1 - (y + 1)*x)).

T(n, k) = (n-1)! * binomial(n, k) for n > 0. - Andrew Howroyd, Oct 11 2017