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A225474 Triangle read by rows, k!*2^k*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

%I A225474 #15 Aug 07 2015 02:48:44
%S A225474 1,1,2,3,8,8,15,46,72,48,105,352,688,768,384,945,3378,7600,11040,9600,
%T A225474 3840,10395,39048,97112,167040,193920,138240,46080,135135,528414,
%U A225474 1418648,2754192,3857280,3736320,2257920,645120,2027025,8196480,23393376,49824768,79892736
%N A225474 Triangle read by rows, k!*2^k*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
%C A225474 The Stirling-Frobenius cycle numbers are defined in A225470.
%H A225474 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedEulerianPolynomials.html">Generalized Eulerian polynomials.</a>
%H A225474 Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">The Stirling-Frobenius numbers.</a>
%F A225474 For a recurrence see the Sage program.
%F A225474 T(n, 0) ~ A001147; T(n, n) ~ A000165; T(n, n-1) ~ A014479.
%F A225474 T(n,k) = A028338(n,k) * A000165(k) = A225475(n,k) * A000079(k) = A161198(n,k) * A000142(k). - _Philippe Deléham_, Jun 25 2015
%e A225474 [n\k][ 0,    1,    2,     3,    4,    5]
%e A225474 [0]    1,
%e A225474 [1]    1,    2,
%e A225474 [2]    3,    8,    8,
%e A225474 [3]   15,   46,   72,    48,
%e A225474 [4]  105,  352,  688,   768,  384,
%e A225474 [5]  945, 3378, 7600, 11040, 9600, 3840.
%t A225474 SFCSO[n_, k_, m_] := SFCSO[n, k, m] = If[k>n || k<0, 0, If[n == 0 && k == 0, 1, m*k*SFCSO[n-1, k-1, m] + (m*n-1)*SFCSO[n-1, k, m]]]; Table[SFCSO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 05 2014, translated from Sage *)
%o A225474 (Sage)
%o A225474 @CachedFunction
%o A225474 def SF_CSO(n, k, m):
%o A225474     if k > n or k < 0 : return 0
%o A225474     if n == 0 and k == 0: return 1
%o A225474     return m*k*SF_CSO(n-1, k-1, m) + (m*n-1)*SF_CSO(n-1, k, m)
%o A225474 for n in (0..8): [SF_CSO(n, k, 2) for k in (0..n)]
%Y A225474 Cf. A028338, A225479 (m=1), A048594, A161198, A225475.
%Y A225474 Cf. A000079, A000142, A000165, A001147, A014479, A028338.
%K A225474 nonn,tabl
%O A225474 0,3
%A A225474 _Peter Luschny_, May 19 2013