login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A164558 Numerators of the n-th term of the binomial transform of the original Bernoulli numbers. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

LINKS

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

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

EXAMPLE

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

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, 38}] (* Jean-Fran├žois Alcover, Aug 08 2012 *)

PROG

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

CROSSREFS

Cf. A027642, A164555, A196838, A196839.

Sequence in context: A107774 A253685 A122478 * A128368 A050089 A282174

Adjacent sequences:  A164555 A164556 A164557 * A164559 A164560 A164561

KEYWORD

sign,frac

AUTHOR

Paul Curtz, Aug 16 2009

EXTENSIONS

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

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified April 10 22:27 EDT 2021. Contains 342856 sequences. (Running on oeis4.)