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A064636
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Number of derangements up to cyclic rotations; permutation siteswap necklaces, with no fixed points (no "zero-throws", i.e., empty hands, if we use the mapping Perm2SiteSwap1 of A060495 and A060498).
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3
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0, 0, 1, 2, 5, 12, 55, 270, 1893, 14864, 133749, 1334970, 14687195, 176214852, 2290820923, 32071104006, 481066907653, 7697064251760, 130850098582189, 2355301661033970, 44750731672347273, 895014631193654828, 18795307257304746591, 413496759611120779902, 9510425471105377569963, 228250211305338670543432
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OFFSET
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0,4
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COMMENTS
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This sequence counts derangements (enumerated by A000166) up to the same automorphism as permutations (enumerated by A000142) are subjected to in A061417.
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LINKS
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FORMULA
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a(n) = Sum_{d|n} (1/n) * Phi(n/d) * Sum_{k=0..d} [ ((n/d)^(d-k)) * (((n/d)-1)^k) * A008290(d, k) ]. (Note: this abbreviated formula supposes that 0^0 = 1. For a practical implementation, see the Maple procedure below.)
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MAPLE
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with(numtheory); A064636 := proc(n) local d, k, s; s := 0; for d in divisors(n) do s := s + (1/n) * phi(n/d) * ( (((n/d)^d)*A000166(d)) + add((((n/d)^(d-k)) * (((n/d)-1)^k) * (A000166(d-k)*binomial(d, k))), k=1..d)); od; RETURN(s); end;
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MATHEMATICA
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Unprotect[Power]; 0^0 = 1; a[n_] := (1/n) DivisorSum[n, EulerPhi[n/#]*Sum[ (n/#)^(# - k)*(n/# - 1)^k*#!*Gamma[# - k + 1, -1]/(E*k!*(# - k)!), {k, 0, #}]&] // FunctionExpand; a[0] = 0; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Mar 06 2016 *)
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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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