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A225475 Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.

%I A225475 #18 Jun 24 2015 08:17:34
%S A225475 1,1,1,3,4,2,15,23,18,6,105,176,172,96,24,945,1689,1900,1380,600,120,
%T A225475 10395,19524,24278,20880,12120,4320,720,135135,264207,354662,344274,
%U A225475 241080,116760,35280,5040,2027025,4098240,5848344,6228096,4993296,2956800,1229760
%N A225475 Triangle read by rows, k!*s_2(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.
%C A225475 The Stirling-Frobenius cycle numbers are defined in A225470.
%H A225475 Vincenzo Librandi, <a href="/A225475/b225475.txt">Rows n = 0..50, flattened</a>
%H A225475 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedEulerianPolynomials.html">Generalized Eulerian polynomials.</a>
%H A225475 Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">The Stirling-Frobenius numbers.</a>
%F A225475 For a recurrence see the Sage program.
%F A225475 T(n, 0) ~ A001147; T(n, 1) ~ A004041.
%F A225475 T(n, n) ~ A000142; T(n, n-1) ~ A001563.
%F A225475 T(n,k) = A028338(n,k)*A000142(k). - _Philippe Deléham_, Jun 24 2015
%e A225475 [n\k][ 0,    1,    2,    3,   4,   5]
%e A225475 [0]    1,
%e A225475 [1]    1,    1,
%e A225475 [2]    3,    4,    2,
%e A225475 [3]   15,   23,   18,    6,
%e A225475 [4]  105,  176,  172,   96,  24,
%e A225475 [5]  945, 1689, 1900, 1380, 600, 120.
%t A225475 SFCO[n_, k_, m_] := SFCO[n, k, m] = If[ k > n || k < 0, Return[0], If[ n == 0 && k == 0, Return[1], Return[ k*SFCO[n - 1, k - 1, m] + (m*n - 1)*SFCO[n - 1, k, m]]]]; Table[ SFCO[n, k, 2], {n, 0, 8}, {k, 0, n}] // Flatten  (* _Jean-François Alcover_, Jul 02 2013, translated from Sage *)
%o A225475 (Sage)
%o A225475 @CachedFunction
%o A225475 def SF_CO(n, k, m):
%o A225475     if k > n or k < 0 : return 0
%o A225475     if n == 0 and k == 0: return 1
%o A225475     return k*SF_CO(n-1, k-1, m) + (m*n-1)*SF_CO(n-1, k, m)
%o A225475 for n in (0..8): [SF_CO(n, k, 2) for k in (0..n)]
%Y A225475 Cf. A028338, A225479 (m=1), A048594.
%K A225475 nonn,tabl
%O A225475 0,4
%A A225475 _Peter Luschny_, May 19 2013