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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

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Last modified December 22 19:34 EST 2014. Contains 252365 sequences.