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%I #17 Aug 03 2017 03:00:16
%S 0,0,1,2,5,12,55,270,1893,14864,133749,1334970,14687195,176214852,
%T 2290820923,32071104006,481066907653,7697064251760,130850098582189,
%U 2355301661033970,44750731672347273,895014631193654828,18795307257304746591,413496759611120779902,9510425471105377569963,228250211305338670543432
%N 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).
%C This sequence counts derangements (enumerated by A000166) up to the same automorphism as permutations (enumerated by A000142) are subjected to in A061417.
%H Juggling Information Service, <a href="http://www.juggling.org/bin/mfs/JIS/help/siteswap/">Site Swap notation</a>
%F 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.)
%p 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;
%t 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 *)
%K nonn
%O 0,4
%A _Antti Karttunen_, Oct 02 2001
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