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A225477
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Triangle read by rows, 3^k*s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
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1
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1, 2, 3, 10, 21, 9, 80, 198, 135, 27, 880, 2418, 2079, 702, 81, 12320, 36492, 36360, 16065, 3240, 243, 209440, 657324, 727596, 382185, 103275, 13851, 729, 4188800, 13774800, 16523892, 9826488, 3212055, 586845, 56133, 2187, 96342400, 329386800, 421373916, 275580900, 103356729, 23133600, 3051594, 218700, 6561
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history;
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OFFSET
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0,2
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COMMENTS
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Triangle T(n,k), read by rows, given by (2, 3, 5, 6, 8, 9, 11, 12, 14, ... (A007494)) DELTA (3, 0, 3, 0, 3, 0, 3, 0, 3, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 15 2015
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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, 6 ]
[0] 1,
[1] 2, 3,
[2] 10, 21, 9,
[3] 80, 198, 135, 27,
[4] 880, 2418, 2079, 702, 81,
[5] 12320, 36492, 36360, 16065, 3240, 243,
[6] 209440, 657324, 727596, 382185, 103275, 13851, 729.
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MATHEMATICA
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s[_][0, 0] = 1; s[m_][n_, k_] /; (k > n || k < 0) = 0; s[m_][n_, k_] := s[m][n, k] = s[m][n - 1, k - 1] + (m*n - 1)*s[m][n - 1, k];
T[n_, k_] := 3^k*s[3][n, k];
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PROG
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(Sage)
@CachedFunction
def SF_CS(n, k, m):
if k > n or k < 0 : return 0
if n == 0 and k == 0: return 1
return m*SF_CS(n-1, k-1, m) + (m*n-1)*SF_CS(n-1, k, m)
for n in (0..8): [SF_CS(n, k, 3) 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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