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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

LINKS

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

FORMULA

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

EXAMPLE

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,

  ...

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

PROG

(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 */

CROSSREFS

Sequence in context: A062833 A006250 A006249 * A019575 A178882 A186195

Adjacent sequences:  A216968 A216969 A216970 * A216972 A216973 A216974

KEYWORD

nonn,nice,tabl

AUTHOR

Geoffrey Critzer, Sep 21 2012

STATUS

approved

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Last modified January 27 10:50 EST 2021. Contains 340465 sequences. (Running on oeis4.)