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 A064636 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). 3
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,4 COMMENTS This sequence counts derangements (enumerated by A000166) up to the same automorphism as permutations (enumerated by A000142) are subjected to in A061417. LINKS Juggling Information Service, Site Swap notation FORMULA 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.) MAPLE 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; MATHEMATICA 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 *) CROSSREFS Sequence in context: A038576 A002358 A083699 * A293023 A145857 A264863 Adjacent sequences:  A064633 A064634 A064635 * A064637 A064638 A064639 KEYWORD nonn AUTHOR Antti Karttunen, Oct 02 2001 STATUS approved

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Last modified December 13 09:26 EST 2018. Contains 318086 sequences. (Running on oeis4.)