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A246615 Number of endofunctions on [n] whose cycle lengths are multiples of 8. 2

%I

%S 1,0,0,0,0,0,0,0,5040,362880,18144000,804988800,34488115200,

%T 1482082842240,65120246231040,2955402450000000,139297568464454400,

%U 6837892496270035200,350034595015623598080,18692451288585606954240,1041114148173066240000000

%N Number of endofunctions on [n] whose cycle lengths are multiples of 8.

%H Alois P. Heinz, <a href="/A246615/b246615.txt">Table of n, a(n) for n = 0..300</a>

%F E.g.f.: 1/(1-LambertW(-x)^8)^(1/8). - _Vaclav Kotesovec_, Sep 01 2014

%F a(n) ~ n^(n-7/16) * 2^(1/16) * sqrt(Pi) / GAMMA(1/16) * (1 - 161 * sqrt(2/n) * GAMMA(1/16) / (768 * GAMMA(9/16))). - _Vaclav Kotesovec_, Sep 01 2014

%p with(combinat):

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

%p `if`(i>n, 0, add(b(n-i*j, i+8)*(i-1)!^j*

%p multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))

%p end:

%p a:= a->add(b(j, 8)*n^(n-j)*binomial(n-1, j-1), j=0..n):

%p seq(a(n), n=0..25);

%t CoefficientList[Series[1/(1-LambertW[-x]^8)^(1/8),{x,0,20}],x] * Range[0,20]! (* _Vaclav Kotesovec_, Sep 01 2014 *)

%Y Column k=8 of A246609.

%K nonn

%O 0,9

%A _Alois P. Heinz_, Aug 31 2014

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Last modified May 19 23:13 EDT 2022. Contains 353847 sequences. (Running on oeis4.)