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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.
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
0,3
COMMENTS
The Stirling-Frobenius cycle numbers are defined in A225470.
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
EXAMPLE
[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.
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 *)
PROG
(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)]
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 19 2013
STATUS
approved