A074881 Triangle T(n,k) giving number of labeled cyclic subgroups of order k in symmetric group S_n, n >= 1, 1 <= k <= g(n), where g(n) = A000793(n) is Landau's function.

1, 1, 1, 1, 3, 1, 1, 9, 4, 3, 1, 25, 10, 15, 6, 10, 1, 75, 40, 90, 36, 120, 1, 231, 175, 420, 126, 735, 120, 126, 105, 1, 763, 616, 2730, 336, 5320, 960, 1260, 1008, 840, 336, 1, 2619, 2884, 15498, 756, 41580, 4320, 11340, 6720, 6804, 7560, 4320, 3024, 2268

Offset: 1

Author

Vladeta Jovovic, Sep 30 2002

Keywords

• nonn
• tabf

Comments

A057731 contains zeros. This sequence contains only positive values of A057731(n,k)/A000010(k). - Alois P. Heinz, Feb 16 2013

Examples

			Triangle begins:
  1;
  1,   1;
  1,   3,   1;
  1,   9,   4,   3;
  1,  25,  10,  15,   6,  10;
  1,  75,  40,  90,  36, 120;
  1, 231, 175, 420, 126, 735, 120, 126, 105;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..42, flattened

Crossrefs

Row sums give A051625.

Cf. A000010, A181949.

Programs

Mathematica

nmax = 10;
T[n_, k_] := n! SeriesCoefficient[O[x]^(n+1) + Sum[MoebiusMu[k/i]*Exp[ Sum[x^j/j, {j, Divisors[i]}]], {i, Divisors[k]}], {x, 0, n}]/ EulerPhi[k];
Table[DeleteCases[Table[T[n, k], {k, 1, 2 nmax}], 0], {n, 1, nmax}] // Flatten (* Jean-François Alcover, Sep 16 2019, after Andrew Howroyd *)

PARI

T(n,k)={n!*polcoeff(sumdiv(k, i, moebius(k/i)*exp(sumdiv(i, j, x^j/j) + O(x*x^n))), n)/eulerphi(k)} \\ Andrew Howroyd, Jul 02 2018

Formula

T(n,k) = A057731(n,k)/A000010(k).