0,11

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).

Peter Luschny, Stirling-Frobenius numbers

Peter Luschny, Generalized Bernoulli numbers.

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)).

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 *)

(SageMath)

@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)]

Cf. A225481, A226157.

Sequence in context: A027641 A164555 A176327 * A215616 A249737 A129205

Adjacent sequences: A226153 A226154 A226155 * A226157 A226158 A226159

sign,frac

Peter Luschny, May 30 2013

approved