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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). 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,11

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).

LINKS

Table of n, a(n) for n=0..38.

Peter Luschny, Stirling-Frobenius numbers

Peter Luschny, Generalized Bernoulli numbers.

EXAMPLE

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)).

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 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)]

CROSSREFS

Cf. A225481, A226157.

Sequence in context: A027641 A164555 A176327 * A215616 A249737 A129205

Adjacent sequences:  A226153 A226154 A226155 * A226157 A226158 A226159

KEYWORD

sign,frac

AUTHOR

Peter Luschny, May 30 2013

STATUS

approved

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)