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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.
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%I #31 Mar 14 2024 15:43:37

%S 1,3,1,9,10,1,27,79,21,1,81,580,310,36,1,243,4141,3990,850,55,1,729,

%T 29230,48031,16740,1895,78,1,2187,205339,557571,299131,52745,3689,105,

%U 1,6561,1439560,6338620,5044536,1301286,137592,6524,136,1

%N 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 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 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 Vincenzo Librandi, <a href="/A225469/b225469.txt">Rows n = 0..50, flattened</a>

%H P. Bala, <a href="/A143395/a143395.pdf">A 3 parameter family of generalized Stirling numbers</a>.

%H 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 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>

%H 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 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 For a recurrence see the Maple program.

%F T(n, 0) ~ A000244; T(n, 1) ~ A016138; T(n, 2) ~ A018054.

%F T(n, n) ~ A000012; T(n, n-1) ~ A014105.

%F From _Wolfdieter Lang_, Apr 13 2017: (Start)

%F E.g.f.: exp(3*z)*exp((x/4)*(exp(4*z -1))). Sheffer triangle (see a comment above).

%F E.g.f. column k: exp(3*x)*(exp(4*x) -1)^k/(4^k*k!), k >= 0 (Sheffer property).

%F O.g.f. column k: x^m/Product_{j=0..k} (1 - (3+4*j)*x), k >= 0.

%F (End)

%e [n\k][ 0, 1, 2, 3, 4, 5, 6]

%e [0] 1,

%e [1] 3, 1,

%e [2] 9, 10, 1,

%e [3] 27, 79, 21, 1,

%e [4] 81, 580, 310, 36, 1,

%e [5] 243, 4141, 3990, 850, 55, 1,

%e [6] 729, 29230, 48031, 16740, 1895, 78, 1.

%p SF_S := proc(n, k, m) option remember;

%p if n = 0 and k = 0 then return(1) fi;

%p if k > n or k < 0 then return(0) fi;

%p SF_S(n-1, k-1, m) + (m*(k+1)-1)*SF_S(n-1, k, m) end:

%p seq(print(seq(SF_S(n, k, 4), k=0..n)), n = 0..5);

%t 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 (Sage)

%o @CachedFunction

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

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

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

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

%o return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))/(factorial(k)*m^k)

%o for n in (0..6): [SF_S(n, k, 4) for k in (0..n)]

%Y Cf. A048993 (m=1), A039755 (m=2), A225468 (m=3).

%Y Cf. Columns: A000244, A016138, A018054.

%K nonn,easy,tabl

%O 0,2

%A _Peter Luschny_, May 16 2013