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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. 2

%I

%S 1,1,1,3,4,2,15,23,18,6,105,176,172,96,24,945,1689,1900,1380,600,120,

%T 10395,19524,24278,20880,12120,4320,720,135135,264207,354662,344274,

%U 241080,116760,35280,5040,2027025,4098240,5848344,6228096,4993296,2956800,1229760

%N 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 The Stirling-Frobenius cycle numbers are defined in A225470.

%H Vincenzo Librandi, <a href="/A225475/b225475.txt">Rows n = 0..50, flattened</a>

%H Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedEulerianPolynomials.html">Generalized Eulerian polynomials.</a>

%H Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">The Stirling-Frobenius numbers.</a>

%F For a recurrence see the Sage program.

%F T(n, 0) ~ A001147; T(n, 1) ~ A004041.

%F T(n, n) ~ A000142; T(n, n-1) ~ A001563.

%F T(n,k) = A028338(n,k)*A000142(k). - _Philippe Deléham_, Jun 24 2015

%e [n\k][ 0, 1, 2, 3, 4, 5]

%e [0] 1,

%e [1] 1, 1,

%e [2] 3, 4, 2,

%e [3] 15, 23, 18, 6,

%e [4] 105, 176, 172, 96, 24,

%e [5] 945, 1689, 1900, 1380, 600, 120.

%t 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 (Sage)

%o @CachedFunction

%o def SF_CO(n, k, m):

%o if k > n or k < 0 : return 0

%o if n == 0 and k == 0: return 1

%o return k*SF_CO(n-1, k-1, m) + (m*n-1)*SF_CO(n-1, k, m)

%o for n in (0..8): [SF_CO(n, k, 2) for k in (0..n)]

%Y Cf. A028338, A225479 (m=1), A048594.

%K nonn,tabl

%O 0,4

%A _Peter Luschny_, May 19 2013

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Last modified August 26 05:50 EDT 2019. Contains 326330 sequences. (Running on oeis4.)