A226156 a(n) = BS(n) * W(n) where BS = Sum_{k=0..n} ((-1)^k*k!/(k+1)) S(n, k) and S(n, k) the Stirling subset numbers A048993(n, k). W(n) = Product_{ p primes <= n+1 such that p divides n+1 or p-1 divides n } = A225481(n).

1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 35, 0, -3617, 0, 43867, 0, -1222277, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -84802531453387, 0, 90219075042845, 0, -26315271553053477373, 0, 38089920879940267

Offset: 0

Author

Peter Luschny, May 30 2013

Keywords

• sign
• frac

Comments

a(n)/A225481(n) is a representation of the Bernoulli numbers. This is case m = 1 of the scaled generalized Bernoulli numbers defined as Sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{m}(n,k) where S_{m}(n,k) are generalized Stirling subset numbers. A225481(n) can be seen as an analog of the Clausen numbers A141056(n). Reduced to lowest terms a(n)/A225481(n) becomes A027641(n)/A027642(n).

Examples

			The numerators of 1/1, -1/2, 1/6, 0/2, -1/30, 0/6, 1/42, 0/2, -1/30, 0/10, 5/66, 0/6, -691/2730, 0/14, 35/30, 0/2, -3617/510, 0/6, 43867/798, ... (the denominators are A225481(n)).
		

Links

• Peter Luschny, Stirling-Frobenius numbers
• Peter Luschny, Generalized Bernoulli numbers.

Crossrefs

Cf. A225481, A226157.

Programs

Mathematica

BS[n_] := Sum[((-1)^k*k!/(k + 1)) StirlingS2[n, k], {k, 0, n}];
W[n_] := Product[If[Divisible[n + 1, p] || Divisible[n, p - 1], p, 1], {p, Prime /@ Range[PrimePi[n + 1]]}];
a[n_] := BS[n] W[n];
Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Jul 08 2019 *)

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 SF_BS(n, m):   # -- The scaled Stirling-Frobenius 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))
def A226156(n):    # -- The numerators of SF_BS(n, 1) relative to A225481 --
    C = mul(filter(lambda p: ((n+1)%p == 0) or (n%(p-1) == 0), primes(n+2)))
    return C*SF_BS(n, 1)
[A226156(n) for n in (0..25)]