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.

1, 3, 1, 9, 10, 1, 27, 79, 21, 1, 81, 580, 310, 36, 1, 243, 4141, 3990, 850, 55, 1, 729, 29230, 48031, 16740, 1895, 78, 1, 2187, 205339, 557571, 299131, 52745, 3689, 105, 1, 6561, 1439560, 6338620, 5044536, 1301286, 137592, 6524, 136, 1

Offset: 0

Author

Peter Luschny, May 16 2013

Keywords

• nonn
• easy
• tabl

Comments

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).

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

Examples

			[n\k][ 0,     1,     2,     3,    4,   5,  6]
[0]    1,
[1]    3,     1,
[2]    9,    10,     1,
[3]   27,    79,    21,     1,
[4]   81,   580,   310,    36,    1,
[5]  243,  4141,  3990,   850,   55,  1,
[6]  729, 29230, 48031, 16740, 1895, 78,  1.
		

Links

• Vincenzo Librandi, Rows n = 0..50, flattened
• P. Bala, A 3 parameter family of generalized Stirling numbers.
• Paweł Hitczenko, A class of polynomial recurrences resulting in (n/log n, n/log^2 n)-asymptotic normality, arXiv:2403.03422 [math.CO], 2024. See pp. 8-9.
• Peter Luschny, Generalized Eulerian polynomials.
• Peter Luschny, The Stirling-Frobenius numbers.
• Shi-Mei Ma, Toufik Mansour, and Matthias Schork, Normal ordering problem and the extensions of the Stirling grammar, Russian Journal of Mathematical Physics, 2014, 21(2), arXiv 1308.0169 p. 12.

Crossrefs

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

Cf. Columns: A000244, A016138, A018054.

Programs

Maple

SF_S := proc(n, k, m) option remember;
if n = 0 and k = 0 then return(1) fi;
if k > n or k < 0 then return(0) fi;
SF_S(n-1, k-1, m) + (m*(k+1)-1)*SF_S(n-1, k, m) end:
seq(print(seq(SF_S(n, k, 4), k=0..n)), n = 0..5);

Mathematica

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 *)

Sage

@CachedFunction
def EulerianNumber(n, k, m) :
    if n == 0: return 1 if k == 0 else 0
    return (m*(n-k)+m-1)*EulerianNumber(n-1, k-1, m) + (m*k+1)*EulerianNumber(n-1, k, m)
def SF_S(n, k, m):
    return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))/(factorial(k)*m^k)
for n in (0..6): [SF_S(n, k, 4) for k in (0..n)]

Formula

T(n, k) = (sum_{j=0..n} binomial(j, n-k)*A_4(n, j)) / (4^k*k!) where A_4(n,j) = A225118.

For a recurrence see the Maple program.

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

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

From Wolfdieter Lang, Apr 13 2017: (Start)

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

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

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

(End)