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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.
3

%I #43 Aug 09 2022 07:04:23

%S 1,2,2,6,18,3,24,156,72,4,120,1520,1260,220,5,720,17310,21000,7020,

%T 600,6,5040,232932,363720,187320,32970,1554,7,40320,3698744,6794256,

%U 4746840,1351840,141288,3920,8,362880,68680656,139241088,121105152,48822480,8625456,573048,9720,9,3628800,1471193370

%N 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.

%C 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.

%C Row sums = n^n.

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

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

%C Conjecture: every entry in row n is divisible by n. - _Jon Perry_, Sep 21 2012

%H Joerg Arndt, <a href="/A216971/b216971.txt">Table of n, a(n) for n = 1..528</a>

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

%F Sum_{k=1..n-1} k * T(n,k) = A001864(n). - _Geoffrey Critzer_, Jan 01 2013

%e Triangle starts:

%e 1,

%e 2, 2,

%e 6, 18, 3,

%e 24, 156, 72, 4,

%e 120, 1520, 1260, 220, 5,

%e 720, 17310, 21000, 7020, 600, 6,

%e 5040, 232932, 363720, 187320, 32970, 1554, 7,

%e ...

%t 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

%o (PARI)

%o N=15; x='x+O('x^N);

%o T=serreverse(x*exp(-x));

%o egf=1/(1-x*exp('y*T)) - 1;

%o v=Vec(serlaplace(egf));

%o { for (n=1, N-1, /* print triangle: */

%o row = Pol( v[n], 'y );

%o row = polrecip( row );

%o print( Vec(row) );

%o ); }

%o /* _Joerg Arndt_, Sep 21 2012 */

%Y Cf. A001864.

%K nonn,nice,tabl

%O 1,2

%A _Geoffrey Critzer_, Sep 21 2012