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A225473 Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

%I A225473 #22 Jul 12 2017 06:11:33
%S A225473 1,3,4,9,40,32,27,316,672,384,81,2320,9920,13824,6144,243,16564,
%T A225473 127680,326400,337920,122880,729,116920,1536992,6428160,11642880,
%U A225473 9584640,2949120,2187,821356,17842272,114866304,324065280,453304320,309657600,82575360,6561
%N A225473 Triangle read by rows, k!*S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
%C A225473 The Stirling-Frobenius subset numbers are defined in A225468 (see also the Sage program).
%H A225473 Vincenzo Librandi, <a href="/A225473/b225473.txt">Rows n = 0..50, flattened</a>
%H A225473 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedEulerianPolynomials.html">Generalized Eulerian polynomials.</a>
%H A225473 Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">The Stirling-Frobenius numbers.</a>
%F A225473 For a recurrence see the Maple program.
%F A225473 T(n, 0) ~ A000244; T(n, 1) ~ A190541; T(n, n) ~ A047053.
%F A225473 From _Wolfdieter Lang_, Jul 12 2017: (Start)
%F A225473 T(n, k) = A225467(n, k)*k! = A225469(n, k)*(4^k*k!), 0 <= k <= n.
%F A225473 T(n, k) = Sum_{m=0..n} binomial(k,m)*(-1)^(k-m)*(3 + 4*m)^n.
%F A225473 Recurrence: T(n, -1) = 0, T(0, 0) = 1, T(n, k) = 0 if n < k and T(n, k) =
%F A225473 4*k*T(n-1, k-1) + (3 + 4*k)*T(n-1, k) for n >= 1, k = 0..n  (see the Maple program).
%F A225473 E.g.f. row polynomials R(n, x) = Sum_{m=0..n} T(n, k)*x^k: exp(3*z)/(1 - x*(exp(4*z) - 1)).
%F A225473 E.g.f. column k: exp(3*x)*(exp(4*x) - 1)^k, k >= 0.
%F A225473 O.g.f. column k: k!*(4*x)^k/Product_{j=0..k} (1 - (3  + 4*j)*x), k >= 0.
%F A225473 (End)
%e A225473 [n\k][0,      1,       2,       3,        4,       5,       6 ]
%e A225473 [0]   1,
%e A225473 [1]   3,      4,
%e A225473 [2]   9,     40,      32,
%e A225473 [3]  27,    316,     672,     384,
%e A225473 [4]  81,   2320,    9920,   13824,     6144,
%e A225473 [5] 243,  16564,  127680,  326400,   337920,  122880,
%e A225473 [6] 729, 116920, 1536992, 6428160, 11642880, 9584640, 2949120.
%p A225473 SF_SO := proc(n, k, m) option remember;
%p A225473 if n = 0 and k = 0 then return(1) fi;
%p A225473 if k > n or k < 0 then return(0) fi;
%p A225473 m*k*SF_SO(n-1, k-1, m) + (m*(k+1)-1)*SF_SO(n-1, k, m) end:
%p A225473 seq(print(seq(SF_SO(n, k, 4), k=0..n)), n = 0..5);
%t A225473 EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = (If[ n == 0, Return[If[k == 0, 1, 0]]]; Return[(m*(n-k)+m-1)*EulerianNumber[n-1, k-1, m] + (m*k+1)*EulerianNumber[n-1, k, m]]); SFSO[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]; Table[ SFSO[n, k, 4], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 29 2013, translated from Sage *)
%o A225473 (Sage)
%o A225473 @CachedFunction
%o A225473 def EulerianNumber(n, k, m) :
%o A225473     if n == 0: return 1 if k == 0 else 0
%o A225473     return (m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m)+ (m*k+1)*EulerianNumber(n-1, k, m)
%o A225473 def SF_SO(n, k, m):
%o A225473     return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))
%o A225473 for n in (0..6): [SF_SO(n, k, 4) for k in (0..n)]
%Y A225473 Cf. A131689 (m=1), A145901 (m=2), A225472 (m=3).
%K A225473 nonn,tabl
%O A225473 0,2
%A A225473 _Peter Luschny_, May 17 2013