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, 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
Examples
[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.
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_] := 4^k*s[4][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, 4) for k in (0..n)]
Formula
For a recurrence see the Sage program.
T(n,k) = 4^k * A225471(n,k). - Philippe Deléham, May 14 2015.
Comments