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A161722
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Generalised Bernoulli numbers B_n(X,0), X a Dirichlet character modulus 8.
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3
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2, -44, 2166, -196888, 28730410, -6148123332, 1813990148894, -705775346640176, 350112935442888018, -215681051222514096220, 161537815119247080938182, -144555133640020128085896264, 152323571317104251881943249786
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,1
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COMMENTS
| Let X be a periodic arithmetical function with period m. The generalised Bernoulli polynomials B_n(X,x) attached to X are defined by means of the generating function
(1)... t*exp(t*x)/(exp(m*t)-1) * sum {r = 0..m-1} X(r)*exp(r*t)
= sum {n = 0..inf} B_n(X,x)*t^n/n!.
The values B_n(X,0) are generalisations of the Bernoulli numbers (case X = 1). For the theory and properties of these polynomials and numbers see [Cohen, Section 9.4]. In the present case, X is chosen to be the Dirichlet character modulus 8 given by
(2)... X(8*n+1) = X(8*n+7) = 1; X(8*n+3) = X(8*n+5) = -1; X(2*n) = 0.
The odd indexed generalised Bernoulli numbers B_(2*n+1)(X,0) vanish. The current sequence lists the even indexed values B_(2*n)(X,0).
The coefficients of the generalised Bernoulli polynomials B_n(X,x) are listed in A151751.
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REFERENCES
| H. Cohen, Number Theory - Volume II: Analytic and Modern Tools, Graduate Texts in Mathematics. Springer-Verlag.
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FORMULA
| (1)... a(n) = (-1)^(n+1)*2*n*A000464(n-1).
The sequence of generalised Bernoulli numbers
(2)... [B_n(X,0)]n>=2 = [2,0,-44,0,2166,0,...]
has the e.g.f.
(3)... t*(exp(t)-exp(3*t)-exp(5*t)+exp(7*t))/(exp(8*t)-1),
which simplifies to
(4)... t*sinh(t)/cosh(2*t) = 2*t^2/2! - 44*t^4/4! + ....
Hence
(5)... B_(2*n)(X,0) = (-1)^(n+1)*2*n*A000464(n-1) and B_(2*n+1)(X,0) = 0.
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MAPLE
| #A161722
with(gfun):
G(x) := x*sinh(x)/cosh(2*x):
coefflist := seriestolist(series(G(x), x, 30)):
seq((2*n)!*coefflist[2*n+1], n = 1..14];
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CROSSREFS
| A000464, A153641, A151751.
Sequence in context: A140795 A161744 A054732 * A054914 A161745 A048566
Adjacent sequences: A161719 A161720 A161721 * A161723 A161724 A161725
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KEYWORD
| easy,sign
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AUTHOR
| Peter Bala (pbala(AT)talktalk.net), Jun 18 2009
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EXTENSIONS
| Cross-reference corrected by Peter Bala (pbala(AT)talktalk.net), Jun 22 2009
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