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

Author

Andrew Howroyd, Sep 28 2019

Keywords

• nonn

Comments

Achiral loops may also be called periodic palindromes.

Examples

			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
		

Links

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

Crossrefs

Row sums of A305540.

Cf. A000670, A005649, A326895.

Programs

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 *)

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)}

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