login
This site is supported by donations to The OEIS Foundation.

 

Logo

The submissions stack has been unacceptably high for several months now. Please voluntarily restrict your submissions and please help with the editing. (We don't want to have to impose further limits.)

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A177762 Beta polynomials (coefficients in descending order, triangle read by rows) 0
1, 1, 1, -1, 1, -2, -2, 1, -3, -3, 5, 1, -4, -4, 16, 16, 1, -5, -5, 35, 35, -61, 1, -6, -6, 64, 64, -272, -272, 1, -7, -7, 105, 105, -791, -791, 1385, 1, -8, -8, 160, 160, -1856, -1856, 7936, 7936, 1, -9, -9, 231, 231, -3801, -3801, 28839, 28839, -50521 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

beta_n(x) = sum_{k=0..n-1} C(n,k)b(-k)(z-1)^(n-k-1) for n > 0 and beta_0(x)=1. Here b(s) = 2*4^(-s)(zeta(s,1/4)-zeta(s,3/4)) where zeta(s,t) denotes the Hurwitz zeta function.

beta_n(0) are the signed Euler numbers 1,1,-1,-2,5,16,-61,... The sign pattern is the same as in the egf. tanh + sech.

LINKS

Table of n, a(n) for n=0..55.

Eric Weisstein's World of Mathematics, Dirichlet Beta Function

Peter Luschny, Swiss-Knife polynomials and Euler numbers, Blog on OEIS

EXAMPLE

1

1

z - 1

z^2 - 2 z - 2

z^3 - 3 z^2 - 3 z + 5

z^4 - 4 z^3 - 4 z^2 + 16 z + 16

z^5 - 5 z^4 - 5 z^3 + 35 z^2 + 35 z - 61

MAPLE

beta := proc(n, z) option remember; local k;

if n = 0 then 1 else add(`if`(k mod 2 = 1, 0,

binomial(n, k)*beta(k, 0)*(z-1)^(n-k-1)), k=0..n-1) fi end:

CROSSREFS

Cf. A000111

Sequence in context: A200779 A023990 A117894 * A109380 A167754 A011020

Adjacent sequences:  A177759 A177760 A177761 * A177763 A177764 A177765

KEYWORD

easy,sign,tabf

AUTHOR

Peter Luschny, May 13 2010

STATUS

approved

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

Content is available under The OEIS End-User License Agreement .

Last modified August 29 03:27 EDT 2015. Contains 261184 sequences.