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A246614 Number of endofunctions on [n] whose cycle lengths are multiples of 7.

%I A246614 #13 Jun 11 2024 01:38:56
%S A246614 1,0,0,0,0,0,0,720,40320,1632960,60480000,2213719200,82771476480,
%T A246614 3211179491520,130247609057280,5542874078742000,247828800444825600,
%U A246614 11643316737494682240,574455837701776343040,29734324379435446505280,1612668074301849600000000
%N A246614 Number of endofunctions on [n] whose cycle lengths are multiples of 7.
%H A246614 Alois P. Heinz, <a href="/A246614/b246614.txt">Table of n, a(n) for n = 0..300</a>
%F A246614 E.g.f.: 1/(1+LambertW(-x)^7)^(1/7). - _Vaclav Kotesovec_, Sep 01 2014
%F A246614 a(n) ~ n^(n-3/7) * 2^(3/7) * (sqrt(Pi) / (7^(1/7) * Gamma(1/14))) * (1 - 10 * sqrt(2/n) * Gamma(1/14) / (49 * Gamma(4/7))). - _Vaclav Kotesovec_, Sep 01 2014
%p A246614 with(combinat):
%p A246614 b:= proc(n, i) option remember; `if`(n=0, 1,
%p A246614       `if`(i>n, 0, add(b(n-i*j, i+7)*(i-1)!^j*
%p A246614       multinomial(n, n-i*j, i$j)/j!, j=0..n/i)))
%p A246614     end:
%p A246614 a:= a->add(b(j, 7)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A246614 seq(a(n), n=0..25);
%t A246614 CoefficientList[Series[1/(1+LambertW[-x]^7)^(1/7),{x,0,20}],x] * Range[0,20]!  (* _Vaclav Kotesovec_, Sep 01 2014 *)
%Y A246614 Column k=7 of A246609.
%K A246614 nonn
%O A246614 0,8
%A A246614 _Alois P. Heinz_, Aug 31 2014