A225480 a(n) = B2(n) * C(n) where B2(n) are generalized Bernoulli numbers and C(n) the Clausen numbers.

1, 0, -2, 0, 14, 0, -62, 0, 254, 0, -5110, 0, 2828954, 0, -114674, 0, 237036478, 0, -11499383114, 0, 183092554714, 0, -3584085584926, 0, 3965530936622474, 0, -573989008898786, 0, 6375197353574922166, 0, -9251189109760413581110, 0, 33111281730973040956798, 0

Offset: 0

Author

Peter Luschny, May 30 2013

Keywords

• sign
• frac

Comments

The Clausen numbers C(n) are T(n, 1) in A160014.

Examples

			The numerators of 1/1, 0/2, -2/6, 0/2, 14/30, 0/2, -62/42, 0/2, 254/30, 0/2, -5110/66, 0/2, 2828954/2730, ... (the denominators are the Clausen numbers).
		

Links

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

Crossrefs

Cf. A141056, A160014.

Programs

Maple

B := (n, m) -> add(add(add(((-1)^(n-v)/(j+1))*binomial(n,k)*binomial(j, v)*(m*v)^k, v = 0..j), j = 0..k), k = 0..n);
C := proc(n) numtheory[divisors](n);map(i->i+1,%);select(isprime,%);mul(i,i=%) end:
A225480 := n -> B(n, 2)*C(n); seq(A225480(n), n = 0..33);

Mathematica

B[n_, m_] := Sum[((-1)^(n - v)/(j + 1))*Binomial[n, k]*Binomial[j, v]*If[k == 0, 1, (m*v)^k], {k, 0, n}, {j, 0, k}, {v, 0, j}];
c[n_] := Denominator[Sum[Boole[PrimeQ[d + 1]]/(d + 1), {d, Divisors[n]}]];
a[n_] := B[n, 2]*c[n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Aug 02 2019, from Maple *)

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 B(n, m):   # The generalized Bernoulli numbers
    return add(add(EulerianNumber(n, j, m)*binomial(j, n - k)
           for j in (0..n))*(-1)^k/(k+1) for k in (0..n))
def A225480(n):
    if n == 0: return 1
    C = mul(filter(lambda s: is_prime(s) , map(lambda i: i+1, divisors(n))))
    return C*B(n, 2)
print([A225480(n) for n in (0..33)])

Formula

Let B(n,m) = sum_{k = 0..n} sum_{j = 0..k} sum_{v = 0..j} ((-1)^(n-v)/(j+1)) *binomial(n,k)*binomial(j,v)*(m*v)^k then a(n) = B(n,2)*A141056(n).

Let B2(n) = sum_{k=0..n} ((-1)^k*k!/(k+1)) S_{2}(n, k) where S_{2}(n, k) the Stirling-Frobenius subset numbers A039755(n, k) then a(n) = B2(n)*A141056(n).