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A246214 Number of endofunctions on [n] where the largest cycle length equals 4. 2
6, 150, 3240, 72030, 1719060, 44520840, 1252364400, 38167414560, 1255558958280, 44404434904830, 1681726757430720, 67953913291104750, 2919509551303952880, 132943540577100047760, 6397727538671302783680, 324511272091351156939200, 17306903935107005765263200 (list; graph; refs; listen; history; text; internal format)
OFFSET

4,1

LINKS

Alois P. Heinz, Table of n, a(n) for n = 4..200

FORMULA

a(n) ~ (4*exp(25/12) - 3*exp(11/6)) * n^(n-1). - Vaclav Kotesovec, Aug 21 2014

MAPLE

with(combinat):

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,

      add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*

      b(n-i*j, i-1), j=0..n/i)))

    end:

A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, min(j, k)), j=0..n):

a:= n-> A(n, 4) -A(n, 3):

seq(a(n), n=4..25);

MATHEMATICA

multinomial[n_, k_List] := n!/Times @@ (k!);

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[(i - 1)!^j multinomial[n, Join[{n - i*j}, Table[i, {j}]]]/j! b[n - i*j, i - 1], {j, 0, n/i}]]];

A[n_, k_] := Sum[Binomial[n-1, j-1] n^(n-j) b[j, Min[j, k]], {j, 0, n}];

a[n_] := A[n, 4] - A[n, 3];

a /@ Range[4, 25] (* Jean-Fran├žois Alcover, Dec 28 2020, after Alois P. Heinz *)

CROSSREFS

Column k=4 of A241981.

Sequence in context: A346694 A070025 A291110 * A065946 A222051 A285747

Adjacent sequences:  A246211 A246212 A246213 * A246215 A246216 A246217

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Aug 19 2014

STATUS

approved

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Last modified January 17 19:55 EST 2022. Contains 350406 sequences. (Running on oeis4.)