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A291115 Number of endofunctions on [n] such that the LCM of their cycle lengths equals nine.

%I A291115 #17 Apr 24 2021 08:45:27
%S A291115 0,0,0,0,0,0,0,0,0,40320,4032000,266112000,15008716800,794060467200,
%T A291115 41179634496000,2142915046272000,113401428940800000,
%U A291115 6150985123214131200,343578020565722342400,19818131438503157760000,1182304993642509574656000,73005714001076187082752000
%N A291115 Number of endofunctions on [n] such that the LCM of their cycle lengths equals nine.
%H A291115 Alois P. Heinz, <a href="/A291115/b291115.txt">Table of n, a(n) for n = 0..387</a>
%F A291115 a(n) ~ (3*exp(13/9) - 2*exp(4/3)) * n^(n-1). - _Vaclav Kotesovec_, Aug 18 2017
%p A291115 b:= proc(n, m) option remember; (k-> `if`(m>k, 0,
%p A291115       `if`(n=0, `if`(m=k, 1, 0), add(b(n-j, ilcm(m, j))
%p A291115        *binomial(n-1, j-1)*(j-1)!, j=1..n))))(9)
%p A291115     end:
%p A291115 a:= n-> add(b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n):
%p A291115 seq(a(n), n=0..22);
%t A291115 Unprotect[Power]; Power[0|0., 0|0.]=1; Protect[Power];b[n_, m_]:=b[n, m]=If[m>9, 0, If[n==0, If[m==9,1, 0], Sum[b[n - j, LCM[m, j]] Binomial[n - 1, j - 1](j - 1)!, {j, n}]]]; Table[Sum[b[j, 1]*n^(n -j) Binomial[n - 1, j - 1], {j, 0, n}], {n, 0, 25}] (* _Indranil Ghosh_, Aug 18 2017 *)
%o A291115 (Python)
%o A291115 from sympy.core.cache import cacheit
%o A291115 from sympy import binomial, lcm, factorial as f
%o A291115 @cacheit
%o A291115 def b(n, m): return 0 if m>9 else (1 if m==9 else 0) if n==0 else sum([b(n - j, lcm(m, j))*binomial(n - 1, j - 1)*f(j - 1) for j in range(1, n + 1)])
%o A291115 def a(n): return sum([b(j, 1)*n**(n - j)*binomial(n - 1, j - 1) for j in range(n + 1)])
%o A291115 print([a(n) for n in range(26)]) # _Indranil Ghosh_, Aug 18 2017
%Y A291115 Column k=9 of A222029.
%K A291115 nonn
%O A291115 0,10
%A A291115 _Alois P. Heinz_, Aug 17 2017