A226157 a(n) = BS2(n) * W(n) where BS2 = sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{2}(n, k) and S_{2}(n, k) are the Stirling-Frobenius subset numbers A039755(n, k). W(n) = product{p primes <= n+1 such that p divides n+1 or p-1 divides n} = A225481(n).

1, 1, -2, -2, 14, 33, -62, -132, 254, 14585, -5110, -313266, 2828954, 38669001, -573370, -404801672, 237036478, 117650567067, -11499383114, -24255028327410, 1281647882998, 8203584532193105, -3584085584926, -418397193140056356, 3965530936622474, 405233976502715850633

Offset

0,3

Comments

a(n)/A225481(n) is case m = 2 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 Stirling-Frobenius subset numbers. A225481(n) can be seen as an analog of the Clausen numbers A141056(n).

Links

Table of n, a(n) for n=0..25.

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)))
def A226157(n):   # The numerators of BS(n, 2) relative to A225481
    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

Cf. A225481, A226156.

Sequence in context: A151353 A151437 A235349 * A264508 A194689 A277556

Adjacent sequences: A226154 A226155 A226156 * A226158 A226159 A226160

Keyword

sign,frac

Author

Peter Luschny, May 30 2013

Status

approved