A225476 Triangle read by rows, k!*2^k*S_2(n, k) where S_m(n, k) are the Stirling-Frobenius subset numbers of order m; n >= 0, k >= 0.

1, 1, 1, 1, 4, 2, 1, 13, 18, 6, 1, 40, 116, 96, 24, 1, 121, 660, 1020, 600, 120, 1, 364, 3542, 9120, 9480, 4320, 720, 1, 1093, 18438, 74466, 121800, 94920, 35280, 5040, 1, 3280, 94376, 576576, 1394064, 1653120, 1028160, 322560, 40320, 1, 9841, 478440, 4319160

Offset: 0

Author

Peter Luschny, May 19 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 ]
[0]   1,
[1]   1,   1,
[2]   1,   4,   2,
[3]   1,  13,  18,    6,
[4]   1,  40, 116,   96,  24,
[5]   1, 121, 660, 1020, 600, 120.
		

Links

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

T(n, 0) ~ A000012; T(n, 1) ~ A003462; T(n, 2) ~ A007798.

T(n, n) ~ A000142; T(n, n-1) ~ A001563.

Alternating row sum ~ A000364 (Euler secant numbers).

Cf. A225468, A131689 (m=1).

Programs

Maple

SF_SSO := 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;
k*SF_SSO(n-1, k-1, m) + (m*(k+1)-1)*SF_SSO(n-1, k, m) end:
seq(print(seq(SF_SSO(n, k, 2), 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]]); SFSSO[n_, k_, m_] := Sum[ EulerianNumber[n, j, m]*Binomial[j, n - k], {j, 0, n}]/m^k; Table[ SFSSO[n, k, 2], {n, 0, 9}, {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_SSO(n, k, m):
    return add(EulerianNumber(n, j, m)*binomial(j, n - k) for j in (0..n))/m^k
for n in (0..6): [SF_SSO(n, k, 2) for k in (0..n)]

Formula

T(n, k) = sum_{j=0..n} A_2(n, j)*binomial(j, n-k), where A_2(n, j) are the generalized Eulerian numbers of order m=2.

For a recurrence see the Maple program.