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A246220 Number of endofunctions on [n] where the largest cycle length equals 10.

%I A246220 #9 Aug 21 2014 05:48:05
%S A246220 362880,43908480,3448811520,228012744960,13954338478080,
%T A246220 827512686000000,48753634065776640,2895879112057451520,
%U A246220 174984885490926551040,10817178515493080290560,686533182382689959116800,44833266187415969387604480,3016487768851293040555130880
%N A246220 Number of endofunctions on [n] where the largest cycle length equals 10.
%C A246220 In general, number of endofunctions on [n] where the largest cycle length equals k is asymptotic to (k*exp(H(k)) - (k-1)*exp(H(k-1))) * n^(n-1), where H(k) is the harmonic number A001008/A002805, k>=1. - _Vaclav Kotesovec_, Aug 21 2014
%H A246220 Alois P. Heinz, <a href="/A246220/b246220.txt">Table of n, a(n) for n = 10..200</a>
%F A246220 a(n) ~ (10*exp(7381/2520) - 9*exp(7129/2520)) * n^(n-1). - _Vaclav Kotesovec_, Aug 21 2014
%p A246220 with(combinat):
%p A246220 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A246220       add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
%p A246220       b(n-i*j, i-1), j=0..n/i)))
%p A246220     end:
%p A246220 A:= (n, k)-> add(binomial(n-1, j-1)*n^(n-j)*b(j, min(j, k)), j=0..n):
%p A246220 a:= n-> A(n, 10) -A(n, 9):
%p A246220 seq(a(n), n=10..25);
%Y A246220 Column k=10 of A241981.
%K A246220 nonn
%O A246220 10,1
%A A246220 _Alois P. Heinz_, Aug 19 2014