A164558 Numerators of the n-th term of the binomial transform of the original Bernoulli numbers.

1, 3, 13, 3, 119, 5, 253, 7, 239, 9, 665, 11, 32069, 13, 91, 15, 4543, 17, 58231, 19, -168011, 21, 857549, 23, -236298571, 25, 8553259, 27, -23749436669, 29, 8615841705665, 31, -7709321024897, 33, 2577687858571, 35, -26315271552984386533, 37, 2929993913841787

Offset: 0

Author

Paul Curtz, Aug 16 2009

Keywords

• sign
• frac

Comments

We start from the sequence A164555(i)/A027642(i) of the "original" Bernoulli numbers, i >= 0, and compute its binomial transform, which is the sequence of fractions 1, 3/2, 13/6, 3, 119/30, 5, 253/42, 7, 239/30, 9, ... The a(n) are the numerators of these fractions.

These fractions are also the successive values of Bernoulli(n,2). - N. J. A. Sloane, Nov 10 2009

(-1)^n*a(n)/A027642, with e.g.f. x/(exp(x)*(exp(x)-1)), gives the alternating row sums of the triangle of coefficients of the Bernoulli polynomials A053382/A053383 or A196838/A196839. - Wolfdieter Lang, Oct 25 2011

Examples

			Numerators of the polynomials b(n,x) at x=1 for n >= 0. The first few are: 1, 1/2 + x, 1/6 + x + x^2, (1/2)*x + (3/2)*x^2 + x^3, -1/30 + x^2 + 2*x^3 + x^4, -(1/6)*x +(5/3)*x^3 + (5/2)*x^4 + x^5, ... - _Peter Luschny_, Aug 18 2018
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..250

Crossrefs

Cf. A027642, A164555, A196838, A196839.

Programs

Magma

A164558:= func< n | Numerator((&+[(-1)^j*Binomial(n,j)*Bernoulli(j): j in [0..n]])) >;
[A164558(n): n in [0..50]]; // G. C. Greubel, Feb 24 2023

Maple

read("transforms") : nmax := 40: a := BINOMIAL([seq(A164555(n)/A027642(n),n=0..nmax)]) : seq( numer(op(n,a)),n=1..nmax+1) ; # R. J. Mathar, Aug 26 2009
A164558 := n -> `if`(type(n, odd) and n > 1, n, numer((-1)^n*bernoulli(n,-1))):
seq(A164558(n), n=0..50); # Peter Luschny, Jun 15 2012, revised Aug 18 2018

Mathematica

a[n_]:= Sum[(-1)^k*Binomial[n, k]*BernoulliB[k], {k,0,n}]//Numerator;
Table[a[n], {n,0,50}] (* Jean-François Alcover, Aug 08 2012 *)

PARI

a(n) = numerator(subst(bernpol(n, x), x, 2)); \\ Michel Marcus, Mar 03 2020

SageMath

def A164558(n): return sum((-1)^j*binomial(n,j)*bernoulli(j) for j in range(n+1)).numerator()
[A164558(n) for n in range(51)] # G. C. Greubel, Feb 24 2023

Formula

E.g.f. for a(n)/A027642: x/(exp(-x)*(1-exp(-x))). - Wolfdieter Lang, Oct 25 2011

Let b_{n}(x) = B_{n}(x) - 2*x*[x^(n-1)]B_{n}(x), then a(n) = numerator(b_{n}(1)). - Peter Luschny, Jun 15 2012

Numerators of the polynomials b(n,x) generated by exp(x*z)*z/(1-exp(-z)) evaluated x=1. b(n,x) are the Bernoulli polynomials B(n,x) with a different sign schema, b(n,x) = (-1)^n*B(n,-x) (see the example section). In other words: a(n) = numerator((-1)^n*Bernoulli(n,-1)). a(n) = n for odd n >= 3. - Peter Luschny, Aug 18 2018

Extensions

Edited and extended by R. J. Mathar, Aug 26 2009