A327868 Number of achiral loops (necklaces or bracelets) of length n with integer entries that cover an initial interval of positive integers.

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

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.

a(2n-1) = A000670(n), a(2n) = A005649(n). - Michael Somos, May 04 2022

Example

The a(4) = 8 achiral loops are:
  1111,
  1122, 1112, 1212, 1222,
  1213, 1232, 1323.
G.f. = 1 + x + 2*x^2 + 3*x^3 + 8*x^4 + 13*x^5 + 44*x^6 + 75*x^7 + ... - Michael Somos, May 04 2022

Mathematica

a[ n_] := If[n < 0, 0, Sum[ k!*(StirlingS2[Quotient[n+1, 2], k] + StirlingS2[Quotient[n+2, 2], k]), {k, 0, n+1}]/2]; (* Michael Somos, May 04 2022 *)
a[ n_] := If[n < 0, 0, With[{m = Quotient[n+1, 2]},
m!*SeriesCoefficient[1/(2 - Exp@x)^Mod[n, 2, 1], {x, 0, m}]]]; (* Michael Somos, May 04 2022 *)

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. A000670, A005649, A326895.

Sequence in context: A129873 A025575 A077055 * A145929 A153865 A369774

Adjacent sequences: A327865 A327866 A327867 * A327869 A327870 A327871

Keyword

nonn

Author

Andrew Howroyd, Sep 28 2019

Status

approved