OFFSET
0,3
COMMENTS
LINKS
Peter Luschny, Stirling-Frobenius numbers
Peter Luschny, Generalized Bernoulli numbers.
EXAMPLE
The numerators of 1/1, 1/2, -2/6, -2/2, 14/30, 33/6, -62/42, -132/2, 254/30, 14585/10, -5110/66, ...(the denominators are A225481(n)).
MATHEMATICA
EulerianNumber[n_, k_, m_] := EulerianNumber[n, k, m] = If[n == 0, If[k == 0, 1 , 0], (m*(n-k) + m - 1)*EulerianNumber[n-1, k-1, m] + (m*k + 1)* EulerianNumber[n-1, k, m]];
BS[n_, m_] := Sum[Sum[EulerianNumber[n, j, m]*Binomial[j, n-k], {j, 0, n}]/ ((-m)^k*(k+1)), {k, 0, n}]
a[n_] := Product[If[Divisible[n+1, p] || Divisible[n, p-1], p, 1], {p, Prime /@ Range @ PrimePi[n+1]}] * BS[n, 2];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Jun 27 2019, from Sage *)
PROG
(Sage)
@CachedFunction
def EulerianNumber(n, k, m) : # The Eulerian numbers
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))
@CachedFunction
def BS(n, m): # The generalized scaled Bernoulli numbers
return (add(add(EulerianNumber(n, j, m)*binomial(j, n - k)
for j in (0..n))/((-m)^k*(k+1)) for k in (0..n)))
C = mul(filter(lambda p: ((n+1)%p == 0) or (n%(p-1) == 0), primes(n+2)))
return C*BS(n, 2)
[A226157(n) for n in (0..25)]
CROSSREFS
KEYWORD
sign,frac
AUTHOR
Peter Luschny, May 30 2013
STATUS
approved