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A160035
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Clausen-normalized numerators of the Bernoulli numbers of order 2.
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1, 0, -1, 0, 3, 0, -5, 0, 7, 0, -45, 0, 7601, 0, -91, 0, 54255, 0, -745739, 0, 3317609, 0, -17944773, 0, 5436374093, 0, -213827575, 0, 641235447783, 0, -249859397004145, 0, 238988952277727
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Let B_n{^(k)}(x) denote the Bernoulli polynomials of order k, defined by the generating function
(t/(exp(t)-1))^k*exp(x*t) = Sum_{n>=0} B_n{^(k)}(x) t^n/n!
Bernoulli numbers of order 1 (defined as B_n{^(1)}(1)) can be regarded as a pair of sequences B1_n = N1_n / D1_n with
N1_n = A027641, D1_n = A141056 (Clausen).
Similarly Bernoulli numbers of order 2 (defined as B_n{^(2)}(1)) can be regarded as a pair of sequences B2_n = N2_n / D2_n with
N2_n = this sequence, D2_n = A141056 (Clausen).
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, Boston, Mass., 1974.
C. Jordan, Calculus of Finite Differences, New York, Chelsea, 1965.
N. E. Noerlund, Vorlesungen Ueber Differenzenrechnung, Berlin, Springer-Verlag, 1924.
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EXAMPLE
| The Clausen-normalized Bernoulli polynomials of order 2 are:
1
2 x - 2
6 x^2 - 12 x + 5
2 x^3 - 6 x^2 + 5 x - 1
30 x^4 - 120 x^3 + 150 x^2 - 60 x + 3
2 x^5 - 10 x^4 + 50/3 x^3 - 10 x^2 + x + 1/3
42 x^6 - 252 x^5 + 525 x^4 - 420 x^3 + 63 x^2 + 42 x - 5
The value of these polynomials at x = 1 gives the sequence.
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MAPLE
| a := proc(n) local g, c, i; g := k -> (t/(exp(t)-1))^k*exp(x*t): c := proc(n) local i; mul(i, i=select(isprime, map(i->i+1, numtheory[divisors](n)))) end: convert(series(g(2), t, n+8), polynom): seq(i!*c(i)*subs(x=1, coeff(%, t, i)), i=0..n) end:
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CROSSREFS
| Cf. A120282, A132094, A100615 and A027643.
Sequence in context: A071649 A049283 A141162 * A051704 A049689 A118657
Adjacent sequences: A160032 A160033 A160034 * A160036 A160037 A160038
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KEYWORD
| sign
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AUTHOR
| Peter Luschny (peter(AT)luschny.de), Apr 30 2009
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