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.

1, 3, 4, 9, 40, 32, 27, 316, 672, 384, 81, 2320, 9920, 13824, 6144, 243, 16564, 127680, 326400, 337920, 122880, 729, 116920, 1536992, 6428160, 11642880, 9584640, 2949120, 2187, 821356, 17842272, 114866304, 324065280, 453304320, 309657600, 82575360, 6561

Offset: 0

Author

Peter Luschny, May 17 2013

Keywords

• nonn
• tabl

Comments

The Stirling-Frobenius subset numbers are defined in A225468 (see also the Sage program).

Examples

			[n\k][0,      1,       2,       3,        4,       5,       6 ]
[0]   1,
[1]   3,      4,
[2]   9,     40,      32,
[3]  27,    316,     672,     384,
[4]  81,   2320,    9920,   13824,     6144,
[5] 243,  16564,  127680,  326400,   337920,  122880,
[6] 729, 116920, 1536992, 6428160, 11642880, 9584640, 2949120.
		

Links

• Vincenzo Librandi, Rows n = 0..50, flattened
• Peter Luschny, Generalized Eulerian polynomials.
• Peter Luschny, The Stirling-Frobenius numbers.

Crossrefs

Cf. A131689 (m=1), A145901 (m=2), A225472 (m=3).

Programs

Maple

SF_SO := 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;
m*k*SF_SO(n-1, k-1, m) + (m*(k+1)-1)*SF_SO(n-1, k, m) end:
seq(print(seq(SF_SO(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]]); 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 *)

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_SO(n, k, m):
    return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))
for n in (0..6): [SF_SO(n, k, 4) for k in (0..n)]

Formula

For a recurrence see the Maple program.

T(n, 0) ~ A000244; T(n, 1) ~ A190541; T(n, n) ~ A047053.

From Wolfdieter Lang, Jul 12 2017: (Start)

T(n, k) = A225467(n, k)*k! = A225469(n, k)*(4^k*k!), 0 <= k <= n.

T(n, k) = Sum_{m=0..n} binomial(k,m)*(-1)^(k-m)*(3 + 4*m)^n.

Recurrence: T(n, -1) = 0, T(0, 0) = 1, T(n, k) = 0 if n < k and T(n, k) =

4*k*T(n-1, k-1) + (3 + 4*k)*T(n-1, k) for n >= 1, k = 0..n (see the Maple program).

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)).

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

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

(End)