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%I #43 Mar 14 2024 15:43:46
%S 1,2,1,4,7,1,8,39,15,1,16,203,159,26,1,32,1031,1475,445,40,1,64,5187,
%T 12831,6370,1005,57,1,128,25999,107835,82901,20440,1974,77,1,256,
%U 130123,888679,1019746,369061,53998,3514,100,1
%N Triangle read by rows, S_3(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. For m = 1 this gives the classical Stirling set numbers A048993. (See the links for details.)
%C From _Peter Bala_, Jan 27 2015: (Start)
%C Exponential Riordan array [ exp(2*z), 1/3*(exp(3*z) - 1)].
%C Triangle equals P * A111577 = P^(-1) * A075498, where P is Pascal's triangle A007318.
%C Triangle of connection constants between the polynomial basis sequences {x^n}n>=0 and { n!*3^n*binomial((x - 2)/3,n) }n>=0. An example is given below.
%C This triangle is the particular case a = 3, b = 0, c = 2 of the triangle of generalized Stirling numbers of the second kind S(a,b,c) defined in the Bala link. (End)
%H Vincenzo Librandi, <a href="/A225468/b225468.txt">Rows n = 0..50, flattened</a>
%H Peter 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 p. 8.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1707.04451">On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers</a>, arXiv:1707.04451 [math.NT], 2017.
%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_3(n, j)) / (3^k*k!) with A_3(n,j) = A225117.
%F For a recurrence see the Maple program.
%F T(n, 0) ~ A000079; T(n, 1) ~ A016127; T(n, 2) ~ A016297; T(n, 3) ~ A025999;
%F T(n, n) ~ A000012; T(n, n-1) ~ A005449; T(n, n-2) ~ A024212.
%F From _Peter Bala_, Jan 27 2015: (Start)
%F T(n,k) = sum {i = 0..n} (-1)^(n+i)*3^(i-k)*binomial(n,i)*Stirling2(i+1,k+1).
%F E.g.f.: exp(2*z)*exp(x/3*(exp(3*z) - 1)) = 1 + (2 + x)*z + (4 + 7*x + x^2)*z^2/2! + ....
%F T(n,k) = 1/(3^k*k!)*sum {j = 0..k} (-1)^(k-j)*binomial(k,j)*(3*j + 2)^n.
%F O.g.f. for n-th diagonal: exp(-2*x/3)*sum {k >= 0} (3*k + 2)^(k + n - 1)*((x/3*exp(-x))^k)/k!.
%F O.g.f. column k: 1/( (1 - 2*x)*(1 - 5*x)...(1 - (3*k + 2)*x ). (End)
%F E.g.f. column k: exp(2*x)*(exp(3*x - 1)/3^k, k >= 0. See the Bala link for the S(3,0,2) exponential Riordan aka Sheffer triangle. - _Wolfdieter Lang_, Apr 10 2017
%e [n\k][ 0, 1, 2, 3, 4, 5, 6]
%e [0] 1,
%e [1] 2, 1,
%e [2] 4, 7, 1,
%e [3] 8, 39, 15, 1,
%e [4] 16, 203, 159, 26, 1,
%e [5] 32, 1031, 1475, 445, 40, 1,
%e [6] 64, 5187, 12831, 6370, 1005, 57, 1.
%e Connection constants: Row 3: [8, 39, 15, 1] so
%e x^3 = 8 + 39*(x - 2) + 15*(x - 2)*(x - 5) + (x - 2)*(x - 5)*(x - 8). - _Peter Bala_, Jan 27 2015
%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, 3), 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, 3], {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, 3) for k in (0..n)]
%Y Cf. A048993 (m=1), A039755 (m=2), A225469 (m=4).
%Y Cf. A075498, A111577. Columns: A000079, A016127, A016297, A025999. A225466, A225472.
%K nonn,easy,tabl
%O 0,2
%A _Peter Luschny_, May 16 2013
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