A216971 Triangle read by rows: T(n,k) is the number of functions f:{1,2,...,n}->{1,2,...,n} that have exactly k nonrecurrent elements mapped to some (one or more) recurrent element. n >= 1, 0 <= k <= n-1.

1, 2, 2, 6, 18, 3, 24, 156, 72, 4, 120, 1520, 1260, 220, 5, 720, 17310, 21000, 7020, 600, 6, 5040, 232932, 363720, 187320, 32970, 1554, 7, 40320, 3698744, 6794256, 4746840, 1351840, 141288, 3920, 8, 362880, 68680656, 139241088, 121105152, 48822480, 8625456, 573048, 9720, 9, 3628800, 1471193370

Offset: 1

Author

Geoffrey Critzer, Sep 21 2012

Keywords

• nonn
• nice
• tabl

Comments

x in {1,2,...,n} is a recurrent element if there is some k such that f^k(x) = x where f^k(x) denotes iterated functional composition. In other words, a recurrent element is in a cycle of the functional digraph.

Row sums = n^n.

First column (k = 0) counts the n! bijective functions.

T(n,n-1) counts the n constant functions.

Conjecture: every entry in row n is divisible by n. - Jon Perry, Sep 21 2012

Examples

			Triangle starts:
     1,
     2,      2,
     6,     18,      3,
    24,    156,     72,      4,
   120,   1520,   1260,    220,      5,
   720,  17310,  21000,   7020,    600,      6,
  5040, 232932, 363720, 187320,  32970,   1554,      7,
  ...
		

Links

• Joerg Arndt, Table of n, a(n) for n = 1..528

Crossrefs

Cf. A001864.

Programs

Mathematica

nn=7;t=Sum[n^(n-1)x^n/n!,{n,1,nn}];f[list_]:=Select[list,#>0&];Drop[Map[f,Range[0,nn]! CoefficientList[Series[1/(1-x Exp[y t]),{x,0,nn}],{x,y}]],1]//Grid

PARI

N=15; x='x+O('x^N);
T=serreverse(x*exp(-x));
egf=1/(1-x*exp('y*T)) - 1;
v=Vec(serlaplace(egf));
{ for (n=1, N-1, /* print triangle: */
    row = Pol( v[n], 'y );
    row = polrecip( row );
    print( Vec(row) );
); }
/* Joerg Arndt, Sep 21 2012 */

Formula

E.g.f.: 1/(1-x*exp(y*T(x))) - 1 where T(x) is the e.g.f. for A000169.

Sum_{k=1..n-1} k * T(n,k) = A001864(n). - Geoffrey Critzer, Jan 01 2013