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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.

%I A225478 #17 Aug 02 2019 13:45:33
%S A225478 1,3,4,21,40,16,231,524,336,64,3465,8784,7136,2304,256,65835,180756,
%T A225478 170720,72320,14080,1024,1514205,4420728,4649584,2346240,613120,79872,
%U A225478 4096,40883535,125416476,143221680,81946816,25939200,4609024,430080,16384,1267389585,4051444896,4941537984,3113238016,1131902464,246636544,31768576,2228224,65536
%N 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.
%C A225478 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.
%H A225478 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedEulerianPolynomials.html">Generalized Eulerian polynomials.</a>
%H A225478 Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">The Stirling-Frobenius numbers.</a>
%F A225478 For a recurrence see the Sage program.
%F A225478 T(n,k) = 4^k * A225471(n,k). - _Philippe Deléham_, May 14 2015.
%e A225478 [n\k][    0,       1,       2,       3,      4,     5,    6 ]
%e A225478 [0]       1,
%e A225478 [1]       3,       4,
%e A225478 [2]      21,      40,      16,
%e A225478 [3]     231,     524,     336,      64,
%e A225478 [4]    3465,    8784,    7136,    2304,    256,
%e A225478 [5]   65835,  180756,  170720,   72320,  14080,  1024,
%e A225478 [6] 1514205, 4420728, 4649584, 2346240, 613120, 79872, 4096.
%t A225478 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 A225478 T[n_, k_] := 4^k*s[4][n, k];
%t A225478 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 02 2019 *)
%o A225478 (Sage)
%o A225478 @CachedFunction
%o A225478 def SF_CS(n, k, m):
%o A225478     if k > n or k < 0 : return 0
%o A225478     if n == 0 and k == 0: return 1
%o A225478     return m*SF_CS(n-1, k-1, m) + (m*n-1)*SF_CS(n-1, k, m)
%o A225478 for n in (0..8): [SF_CS(n, k, 4) for k in (0..n)]
%Y A225478 T(n, 0) ~ A008545; T(n, n) ~ A000302; T(n, n-1) ~ A002700.
%Y A225478 row sums ~ A034176; alternating row sums ~ A008545.
%Y A225478 Cf. A225471, A132393 (m=1), A028338 (m=2), A225477 (m=3).
%K A225478 nonn,tabl
%O A225478 0,2
%A A225478 _Peter Luschny_, May 17 2013