OFFSET
0,2
COMMENTS
The Stirling-Frobenius subset numbers are defined in A225468 (see also the Sage program).
LINKS
Vincenzo Librandi, Rows n = 0..50, flattened
Peter Luschny, Generalized Eulerian polynomials.
Peter Luschny, The Stirling-Frobenius numbers.
FORMULA
For a recurrence see the Maple program.
From Wolfdieter Lang, Jul 12 2017: (Start)
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)
EXAMPLE
[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.
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 *)
PROG
(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)]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, May 17 2013
STATUS
approved