A327878 Irregular triangle read by rows: T(n,k) is the number of primitive (period n) periodic palindromes using exactly k different symbols, 1 <= k <= 1 + floor(n/2).

1, 0, 1, 0, 2, 0, 3, 3, 0, 6, 6, 0, 7, 21, 12, 0, 14, 36, 24, 0, 18, 90, 132, 60, 0, 28, 150, 240, 120, 0, 39, 339, 900, 960, 360, 0, 62, 540, 1560, 1800, 720, 0, 81, 1149, 4968, 9300, 7920, 2520, 0, 126, 1806, 8400, 16800, 15120, 5040, 0, 175, 3765, 24588, 71400, 103320, 73080, 20160

Offset

1,5

Comments

Primitive periodic palindromes may also be called achiral Lyndon words.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..2600

Formula

T(n,k) = Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*A284856(n,j).

Column k is the Moebius transform of column k of A305540.

Example

Triangle begins:
  1;
  0,   1;
  0,   2;
  0,   3,    3;
  0,   6,    6;
  0,   7,   21,    12;
  0,  14,   36,    24;
  0,  18,   90,   132,    60;
  0,  28,  150,   240,   120;
  0,  39,  339,   900,   960,    360;
  0,  62,  540,  1560,  1800,    720;
  0,  81, 1149,  4968,  9300,   7920,  2520;
  0, 126, 1806,  8400, 16800,  15120,  5040;
  0, 175, 3765, 24588, 71400, 103320, 73080, 20160;
  ...

Prog

(PARI) T(n, k) = {sumdiv(n, d, moebius(n/d) * k! * (stirling((d+1)\2, k, 2) + stirling(d\2+1, k, 2)))/2}

Crossrefs

Columns k=2..6 are A056498, A056499, A056500, A056501, A056502.

Row sums are A327879.

Cf. A284856, A305540, A327873.

Sequence in context: A209693 A154344 A134409 * A337841 A298605 A180013

Adjacent sequences: A327875 A327876 A327877 * A327879 A327880 A327881

Keyword

nonn,tabf

Author

Andrew Howroyd, Sep 28 2019

Status

approved