A157817 Numerator of Bernoulli(n, 1/4).

1, -1, -1, 3, 7, -25, -31, 427, 127, -12465, -2555, 555731, 1414477, -35135945, -57337, 2990414715, 118518239, -329655706465, -5749691557, 45692713833379, 91546277357, -7777794952988025, -1792042792463, 1595024111042171723, 1982765468311237, -387863354088927172625

Offset: 0

Author

N. J. A. Sloane, Nov 08 2009

Keywords

• sign
• easy
• frac

Comments

From Wolfdieter Lang, Apr 28 2017: (Start)

The rationals r(n) = Sum_{k=0..n} ((-1)^k / (k+1))*A285061(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A225473(n, k) define generalized Bernoulli numbers, named B[4,1](n), in terms of the generalized Stirling2 numbers S2[4,1]. The numerators of r(n) are a(n) and the denominators A141459(n). r(n) = B[4,1](n) = 4^n*B(n, 1/4) with the Bernoulli polynomials B(n, x) = Bernoulli(n, x) from A196838/A196839 or A053382/A053383.

The generalized Bernoulli numbers B[4,3](n) = Sum_{k=0..n} ((-1)^k/(k+1))* A225467(n, k)*k! = Sum_{k=0..n} ((-1)^k/(k+1))*A225473(n, k) satisfy

B[4,3](n) = 4^n*B(n, 3/4) = (-1)^n*B[4,1](n). They have numerators (-1)^n*a(n) and also denominators A141459(n). (End)

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..250

Crossrefs

For denominators see A157818 and A141459.

Programs

Mathematica

Table[Numerator[BernoulliB[n, 1/4]], {n, 0, 50}] (* Vincenzo Librandi, Mar 16 2014 *)

Formula

From Wolfdieter Lang, Apr 28 2017: (Start)

a(n) = numerator(Bernoulli(n, 1/4)) with denominator A157818(n) (see the name).

a(n) = numerator(4^n*Bernoulli(n, 1/4)) with denominator A141459(n) = A157818(n)/4^n.

a(n)*(-1)^n = numerator(4^n*Bernoulli(n, 3/4)) with denominator A141459(n).

(End)