A225474 Triangle read by rows, k!*2^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, 2, 3, 8, 8, 15, 46, 72, 48, 105, 352, 688, 768, 384, 945, 3378, 7600, 11040, 9600, 3840, 10395, 39048, 97112, 167040, 193920, 138240, 46080, 135135, 528414, 1418648, 2754192, 3857280, 3736320, 2257920, 645120, 2027025, 8196480, 23393376, 49824768, 79892736

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,    2,
[2]    3,    8,    8,
[3]   15,   46,   72,    48,
[4]  105,  352,  688,   768,  384,
[5]  945, 3378, 7600, 11040, 9600, 3840.
		

Links

• Peter Luschny, Generalized Eulerian polynomials.
• Peter Luschny, The Stirling-Frobenius numbers.

Crossrefs

Cf. A028338, A225479 (m=1), A048594, A161198, A225475.

Cf. A000079, A000142, A000165, A001147, A014479, A028338.

Programs

Mathematica

SFCSO[n_, k_, m_] := SFCSO[n, k, m] = If[k>n || k<0, 0, If[n == 0 && k == 0, 1, m*k*SFCSO[n-1, k-1, m] + (m*n-1)*SFCSO[n-1, k, m]]]; Table[SFCSO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 05 2014, translated from Sage *)

Sage

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

Formula

For a recurrence see the Sage program.

T(n, 0) ~ A001147; T(n, n) ~ A000165; T(n, n-1) ~ A014479.

T(n,k) = A028338(n,k) * A000165(k) = A225475(n,k) * A000079(k) = A161198(n,k) * A000142(k). - Philippe Deléham, Jun 25 2015