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A225478
Triangle read by rows, 4^k*s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
1
1, 3, 4, 21, 40, 16, 231, 524, 336, 64, 3465, 8784, 7136, 2304, 256, 65835, 180756, 170720, 72320, 14080, 1024, 1514205, 4420728, 4649584, 2346240, 613120, 79872, 4096, 40883535, 125416476, 143221680, 81946816, 25939200, 4609024, 430080, 16384, 1267389585, 4051444896, 4941537984, 3113238016, 1131902464, 246636544, 31768576, 2228224, 65536
OFFSET
0,2
COMMENTS
Triangle T(n,k), read by rows, given by (3, 4, 7, 8, 11, 12, 15, 16, ... (A014601)) DELTA (4, 0, 4, 0, 4, 0, 4, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, May 14 2015.
FORMULA
For a recurrence see the Sage program.
T(n,k) = 4^k * A225471(n,k). - Philippe Deléham, May 14 2015.
EXAMPLE
[n\k][ 0, 1, 2, 3, 4, 5, 6 ]
[0] 1,
[1] 3, 4,
[2] 21, 40, 16,
[3] 231, 524, 336, 64,
[4] 3465, 8784, 7136, 2304, 256,
[5] 65835, 180756, 170720, 72320, 14080, 1024,
[6] 1514205, 4420728, 4649584, 2346240, 613120, 79872, 4096.
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_] := 4^k*s[4][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, 4) for k in (0..n)]
CROSSREFS
T(n, 0) ~ A008545; T(n, n) ~ A000302; T(n, n-1) ~ A002700.
row sums ~ A034176; alternating row sums ~ A008545.
Cf. A225471, A132393 (m=1), A028338 (m=2), A225477 (m=3).
Sequence in context: A339482 A359883 A032830 * A254884 A034475 A156173
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 17 2013
STATUS
approved