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A225469 Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

%I A225469 #31 Mar 14 2024 15:43:37
%S A225469 1,3,1,9,10,1,27,79,21,1,81,580,310,36,1,243,4141,3990,850,55,1,729,
%T A225469 29230,48031,16740,1895,78,1,2187,205339,557571,299131,52745,3689,105,
%U A225469 1,6561,1439560,6338620,5044536,1301286,137592,6524,136,1
%N A225469 Triangle read by rows, S_4(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.
%C A225469 The definition of the Stirling-Frobenius subset numbers: S_m(n, k) = (sum_{j=0..n} binomial(j, n-k)*A_m(n, j)) / (m^k*k!) where A_m(n, j) are the generalized Eulerian numbers (see the links for details).
%C A225469 This is the Sheffer triangle (exp(3*x),(1/4)*(exp(4*x -1))). See the P. Bala link where this is called exponential Riordan array S_{(4,0,3)}. - _Wolfdieter Lang_, Apr 13 2017
%H A225469 Vincenzo Librandi, <a href="/A225469/b225469.txt">Rows n = 0..50, flattened</a>
%H A225469 P. Bala, <a href="/A143395/a143395.pdf">A 3 parameter family of generalized Stirling numbers</a>.
%H A225469 Paweł Hitczenko, <a href="https://arxiv.org/abs/2403.03422">A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality</a>, arXiv:2403.03422 [math.CO], 2024. See pp. 8-9.
%H A225469 Peter Luschny, <a href="http://www.luschny.de/math/euler/GeneralizedEulerianPolynomials.html">Generalized Eulerian polynomials.</a>
%H A225469 Peter Luschny, <a href="http://www.luschny.de/math/euler/StirlingFrobeniusNumbers.html">The Stirling-Frobenius numbers.</a>
%H A225469 Shi-Mei Ma, Toufik Mansour, and Matthias Schork, <a href="http://arxiv.org/abs/1308.0169">Normal ordering problem and the extensions of the Stirling grammar</a>, Russian Journal of Mathematical Physics, 2014, 21(2), arXiv 1308.0169 p. 12.
%F A225469 T(n, k) = (sum_{j=0..n} binomial(j, n-k)*A_4(n, j)) / (4^k*k!) where A_4(n,j) = A225118.
%F A225469 For a recurrence see the Maple program.
%F A225469 T(n, 0) ~ A000244; T(n, 1) ~ A016138; T(n, 2) ~ A018054.
%F A225469 T(n, n) ~ A000012; T(n, n-1) ~ A014105.
%F A225469 From _Wolfdieter Lang_, Apr 13 2017: (Start)
%F A225469 E.g.f.: exp(3*z)*exp((x/4)*(exp(4*z -1))). Sheffer triangle (see a comment above).
%F A225469 E.g.f. column k: exp(3*x)*(exp(4*x) -1)^k/(4^k*k!), k >= 0 (Sheffer property).
%F A225469 O.g.f. column k: x^m/Product_{j=0..k} (1 - (3+4*j)*x), k >= 0.
%F A225469 (End)
%e A225469 [n\k][ 0,     1,     2,     3,    4,   5,  6]
%e A225469 [0]    1,
%e A225469 [1]    3,     1,
%e A225469 [2]    9,    10,     1,
%e A225469 [3]   27,    79,    21,     1,
%e A225469 [4]   81,   580,   310,    36,    1,
%e A225469 [5]  243,  4141,  3990,   850,   55,  1,
%e A225469 [6]  729, 29230, 48031, 16740, 1895, 78,  1.
%p A225469 SF_S := proc(n, k, m) option remember;
%p A225469 if n = 0 and k = 0 then return(1) fi;
%p A225469 if k > n or k < 0 then return(0) fi;
%p A225469 SF_S(n-1, k-1, m) + (m*(k+1)-1)*SF_S(n-1, k, m) end:
%p A225469 seq(print(seq(SF_S(n, k, 4), k=0..n)), n = 0..5);
%t A225469 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]]); SFS[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/(k!*m^k); Table[ SFS[n, k, 4], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 29 2013, translated from Sage *)
%o A225469 (Sage)
%o A225469 @CachedFunction
%o A225469 def EulerianNumber(n, k, m) :
%o A225469     if n == 0: return 1 if k == 0 else 0
%o A225469     return (m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m) + (m*k+1)*EulerianNumber(n-1, k, m)
%o A225469 def SF_S(n, k, m):
%o A225469     return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))/(factorial(k)*m^k)
%o A225469 for n in (0..6): [SF_S(n, k, 4) for k in (0..n)]
%Y A225469 Cf. A048993 (m=1), A039755 (m=2), A225468 (m=3).
%Y A225469 Cf. Columns: A000244, A016138, A018054.
%K A225469 nonn,easy,tabl
%O A225469 0,2
%A A225469 _Peter Luschny_, May 16 2013