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).

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

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

Table of n, a(n) for n=0..25.

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 * A355991 A376148 A347598

Adjacent sequences: A064633 A064634 A064635 * A064637 A064638 A064639

Keyword

nonn

Author

Antti Karttunen, Oct 02 2001

Status

approved