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, 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
Examples
[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.
Links
- Peter Luschny, Generalized Eulerian polynomials.
- Peter Luschny, The Stirling-Frobenius numbers.
Crossrefs
Programs
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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 *)
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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)]
Formula
For a recurrence see the Sage program.
T(n,k) = 3^k * A225470(n,k). - Philippe Deléham, May 14 2015.
Comments