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A225477
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.
1
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
OFFSET
0,2
COMMENTS
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
FORMULA
For a recurrence see the Sage program.
T(n,k) = 3^k * A225470(n,k). - Philippe Deléham, May 14 2015.
EXAMPLE
[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.
MATHEMATICA
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];
Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Aug 02 2019 *)
PROG
(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)]
CROSSREFS
T(n, 0) ~ A008544; T(n, n) ~ A000244.
row sums ~ A034000; alternating row sums ~ A008544.
Cf. A225470, A132393 (m=1), A028338 (m=2), A225478 (m=4).
Sequence in context: A252865 A252868 A352394 * A079161 A069565 A139694
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 17 2013
STATUS
approved