A327873 Irregular triangle read by rows: T(n,k) is the number of length n primitive (aperiodic) palindromes using exactly k different symbols, 1 <= k <= ceiling(n/2).

1, 0, 0, 2, 0, 2, 0, 6, 6, 0, 4, 6, 0, 14, 36, 24, 0, 12, 36, 24, 0, 28, 150, 240, 120, 0, 24, 144, 240, 120, 0, 62, 540, 1560, 1800, 720, 0, 54, 534, 1560, 1800, 720, 0, 126, 1806, 8400, 16800, 15120, 5040, 0, 112, 1770, 8376, 16800, 15120, 5040

Offset

1,4

Links

Andrew Howroyd, Table of n, a(n) for n = 1..2550

Formula

T(n,k) = Sum_{j=1..k} (-1)^(k-j)*binomial(k,j)*A284823(n,j).

T(n,k) = Sum_{d|n} mu(n/d)*k!*Stirling2(ceiling(d/2), k).

Example

Triangle begins:
  1;
  0;
  0,   2;
  0,   2;
  0,   6,    6;
  0,   4,    6;
  0,  14,   36,   24;
  0,  12,   36,   24;
  0,  28,  150,  240,   120;
  0,  24,  144,  240,   120;
  0,  62,  540, 1560,  1800,   720;
  0,  54,  534, 1560,  1800,   720;
  0, 126, 1806, 8400, 16800, 15120, 5040;
  0, 112, 1770, 8376, 16800, 15120, 5040;
  ...

Prog

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

Crossrefs

Columns k=2..6 are A056463, A056464, A056465, A056466, A056467.

Row sums are A327874.

Cf. A284823, A327878.

Sequence in context: A175950 A339423 A066285 * A136665 A047765 A068463

Adjacent sequences: A327870 A327871 A327872 * A327874 A327875 A327876

Keyword

nonn,tabf

Author

Andrew Howroyd, Sep 28 2019

Status

approved