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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.

%I A225477 #17 Aug 02 2019 13:45:27
%S A225477 1,2,3,10,21,9,80,198,135,27,880,2418,2079,702,81,12320,36492,36360,
%T A225477 16065,3240,243,209440,657324,727596,382185,103275,13851,729,4188800,
%U A225477 13774800,16523892,9826488,3212055,586845,56133,2187,96342400,329386800,421373916,275580900,103356729,23133600,3051594,218700,6561
%N 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.
%C A225477 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
%H A225477 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedEulerianPolynomials.html">Generalized Eulerian polynomials.</a>
%H A225477 Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">The Stirling-Frobenius numbers.</a>
%F A225477 For a recurrence see the Sage program.
%F A225477 T(n,k) = 3^k * A225470(n,k). - _Philippe Deléham_, May 14 2015.
%e A225477 [n\k][   0,      1,      2,      3,      4,     5,   6 ]
%e A225477 [0]      1,
%e A225477 [1]      2,      3,
%e A225477 [2]     10,     21,      9,
%e A225477 [3]     80,    198,    135,     27,
%e A225477 [4]    880,   2418,   2079,    702,     81,
%e A225477 [5]  12320,  36492,  36360,  16065,   3240,   243,
%e A225477 [6] 209440, 657324, 727596, 382185, 103275, 13851, 729.
%t A225477 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 A225477 T[n_, k_] := 3^k*s[3][n, k];
%t A225477 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 02 2019 *)
%o A225477 (Sage)
%o A225477 @CachedFunction
%o A225477 def SF_CS(n, k, m):
%o A225477     if k > n or k < 0 : return 0
%o A225477     if n == 0 and k == 0: return 1
%o A225477     return m*SF_CS(n-1, k-1, m) + (m*n-1)*SF_CS(n-1, k, m)
%o A225477 for n in (0..8): [SF_CS(n, k, 3) for k in (0..n)]
%Y A225477 T(n, 0) ~ A008544; T(n, n) ~ A000244.
%Y A225477 row sums ~ A034000; alternating row sums ~ A008544.
%Y A225477 Cf. A225470, A132393 (m=1), A028338 (m=2), A225478 (m=4).
%K A225477 nonn,tabl
%O A225477 0,2
%A A225477 _Peter Luschny_, May 17 2013