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A291115
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Number of endofunctions on [n] such that the LCM of their cycle lengths equals nine.
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2
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0, 0, 0, 0, 0, 0, 0, 0, 0, 40320, 4032000, 266112000, 15008716800, 794060467200, 41179634496000, 2142915046272000, 113401428940800000, 6150985123214131200, 343578020565722342400, 19818131438503157760000, 1182304993642509574656000, 73005714001076187082752000
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OFFSET
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0,10
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LINKS
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FORMULA
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MAPLE
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b:= proc(n, m) option remember; (k-> `if`(m>k, 0,
`if`(n=0, `if`(m=k, 1, 0), add(b(n-j, ilcm(m, j))
*binomial(n-1, j-1)*(j-1)!, j=1..n))))(9)
end:
a:= n-> add(b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..22);
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MATHEMATICA
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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 *)
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PROG
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(Python)
from sympy.core.cache import cacheit
from sympy import binomial, lcm, factorial as f
@cacheit
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)])
def a(n): return sum([b(j, 1)*n**(n - j)*binomial(n - 1, j - 1) for j in range(n + 1)])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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