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A225475
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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.
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2
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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
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OFFSET
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0,4
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COMMENTS
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The Stirling-Frobenius cycle numbers are defined in A225470.
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LINKS
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FORMULA
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For a recurrence see the Sage program.
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EXAMPLE
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[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.
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MATHEMATICA
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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 *)
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PROG
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(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)]
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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