A225475 Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

1, 1, 1, 3, 4, 2, 15, 23, 18, 6, 105, 176, 172, 96, 24, 945, 1689, 1900, 1380, 600, 120, 10395, 19524, 24278, 20880, 12120, 4320, 720, 135135, 264207, 354662, 344274, 241080, 116760, 35280, 5040, 2027025, 4098240, 5848344, 6228096, 4993296, 2956800, 1229760

Offset: 0

Author

Peter Luschny, May 19 2013

Keywords

• nonn
• tabl

Comments

The Stirling-Frobenius cycle numbers are defined in A225470.

Examples

			[n\k][ 0,    1,    2,    3,   4,   5]
[0]    1,
[1]    1,    1,
[2]    3,    4,    2,
[3]   15,   23,   18,    6,
[4]  105,  176,  172,   96,  24,
[5]  945, 1689, 1900, 1380, 600, 120.
		

Links

• Vincenzo Librandi, Rows n = 0..50, flattened
• Peter Luschny, Generalized Eulerian polynomials.
• Peter Luschny, The Stirling-Frobenius numbers.

Crossrefs

Cf. A028338, A225479 (m=1), A048594.

Programs

Mathematica

SFCO[n_, k_, m_] := SFCO[n, k, m] = If[ k > n || k < 0, Return[0], If[ n == 0 && k == 0, Return[1], Return[ k*SFCO[n - 1, k - 1, m] + (m*n - 1)*SFCO[n - 1, k, m]]]]; Table[ SFCO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten  (* Jean-François Alcover, Jul 02 2013, translated from Sage *)

Sage

@CachedFunction
def SF_CO(n, k, m):
    if k > n or k < 0 : return 0
    if n == 0 and k == 0: return 1
    return k*SF_CO(n-1, k-1, m) + (m*n-1)*SF_CO(n-1, k, m)
for n in (0..8): [SF_CO(n, k, 2) for k in (0..n)]

Formula

For a recurrence see the Sage program.

T(n, 0) ~ A001147; T(n, 1) ~ A004041.

T(n, n) ~ A000142; T(n, n-1) ~ A001563.

T(n,k) = A028338(n,k)*A000142(k). - Philippe Deléham, Jun 24 2015