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A327868 Number of achiral loops (necklaces or bracelets) of length n with integer entries that cover an initial interval of positive integers. 2
1, 1, 2, 3, 8, 13, 44, 75, 308, 541, 2612, 4683, 25988, 47293, 296564, 545835, 3816548, 7087261, 54667412, 102247563, 862440068, 1622632573, 14857100084, 28091567595, 277474957988, 526858348381, 5584100659412, 10641342970443, 120462266974148, 230283190977853 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Achiral loops may also be called periodic palindromes.

LINKS

Andrew Howroyd, Table of n, a(n) for n = 0..200

FORMULA

a(n) = (1/2)*Sum_{k=0..n} k!*(Stirling2(floor((n+1)/2), k) + Stirling2(ceiling((n+1)/2), k)) for n > 0.

EXAMPLE

The a(4) = 8 achiral loops are:

  1111,

  1122, 1112, 1212, 1222,

  1213, 1232, 1323.

PROG

(PARI) a(n)={if(n<1, n==0, sum(k=0, n, k!*(stirling((n+1)\2, k, 2)+stirling(n\2+1, k, 2)))/2)}

CROSSREFS

Row sums of A305540.

Cf. A326895.

Sequence in context: A129873 A025575 A077055 * A145929 A153865 A022951

Adjacent sequences:  A327865 A327866 A327867 * A327869 A327870 A327871

KEYWORD

nonn

AUTHOR

Andrew Howroyd, Sep 28 2019

STATUS

approved

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Last modified January 20 19:53 EST 2020. Contains 331096 sequences. (Running on oeis4.)