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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.

%I A219570 #10 Oct 11 2017 21:58:32
%S A219570 0,1,1,1,2,1,2,6,6,2,6,24,36,24,6,24,120,240,240,120,24,120,720,1800,
%T A219570 2400,1800,720,120,720,5040,15120,25200,25200,15120,5040,720,5040,
%U A219570 40320,141120,282240,352800,282240,141120,40320,5040,40320,362880,1451520,3386880,5080320,5080320,3386880,1451520,362880,40320
%N 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.
%C A219570 Row sums are A066318.
%H A219570 Andrew Howroyd, <a href="/A219570/b219570.txt">Table of n, a(n) for n = 0..1274</a>
%F A219570 E.g.f.: log(1/(1 - (y + 1)*x)).
%F A219570 T(n, k) = (n-1)! * binomial(n, k) for n > 0. - _Andrew Howroyd_, Oct 11 2017
%e A219570 0;
%e A219570 1,      1;
%e A219570 1,      2,     1;
%e A219570 2,      6,     6,     2;
%e A219570 6,     24,    36,    24,     6;
%e A219570 24,   120,   240,   240,   120,    24;
%e A219570 120,  720,  1800,  2400,  1800,   720,  120;
%e A219570 720, 5040, 15120, 25200, 25200, 15120, 5040, 720;
%t A219570 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
%o A219570 (PARI) T(n, k) = if(n>0, (n-1)! * binomial(n, k)); \\ _Andrew Howroyd_, Oct 11 2017
%K A219570 nonn,tabl
%O A219570 0,5
%A A219570 _Geoffrey Critzer_, Nov 23 2012