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A000367
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Numerators of Bernoulli numbers B_2n.
(Formerly M4039 N1677)
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93
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1, 1, -1, 1, -1, 5, -691, 7, -3617, 43867, -174611, 854513, -236364091, 8553103, -23749461029, 8615841276005, -7709321041217, 2577687858367, -26315271553053477373, 2929993913841559, -261082718496449122051
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OFFSET
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0,6
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REFERENCES
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.
Richard P. Brent and David Harvey, Fast computation of Bernoulli, Tangent and Secant numbers, Arxiv preprint arXiv:1108.0286, 2011
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
H. T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 230.
G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.
H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.
F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 330.
H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zhi-Hong Sun, Congruences involving Bernoulli polynomials, Discr. Math., 308 (2007), 71-112.
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LINKS
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_Simon Plouffe_, Table of n, a(n) for n = 0..249 [taken from link below]
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles.
J. Butcher, Some applications of Bernoulli numbers
C. K. Caldwell, The Prime Glossary, Bernoulli number
R. Jovanovic, Bernoulli numbers and the Pascal triangle
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
B. C. Kellner, On irregular prime power divisors of the Bernoulli numbers
B. C. Kellner, The structure of Bernoulli numbers
C. Lin and L. Zhipeng, On Bernoulli numbers and its properties
S. O. S. Math, Bernoulli and Euler Numbers
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Niels Nielsen, Traite Elementaire des Nombres de Bernoulli, Gauthier-Villars, 1923, pp. 398.
_Simon Plouffe_, The 250,000-th Bernoulli Number
_Simon Plouffe_, The First 498 Bernoulli numbers [Project Gutenberg Etext]
S. Ramanujan, Some Properties of Bernoulli's Numbers
S. S. Wagstaff, Prime factors of the absolute values of Bernoulli numerators
Eric Weisstein's World of Mathematics, Bernoulli Number.
Wikipedia, Bernoulli number
Index entries for sequences related to Bernoulli numbers.
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FORMULA
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E.g.f: x/(exp(x) - 1); take numerators of even powers.
B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k=1..inf} 1/k^(2n) (gives asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} ~ (-1)^(n-1)*2*(2*n)!/(2*Pi)^(2*n).
If n>=3 is prime, then 12*|a((n+1)/2)|==(-1)^((n-1)/2)*A002445((n+1)/2) (mod n). [From Vladimir Shevelev, Sep 04 2010]
G.f. for A000367(n)/A002445(n) :
a(n)= -I*(2*n)!/(Pi*(1-2*n))*integral(log(1-1/t)^(1-2*n) dt, t=0..1) - [Gerry Martens, May 17 2011]
a(n) = numerator((-1)^(n+1)*(2*Pi)^(-2*n)*(2*n)!*Li_{2*n}(1)) for n > 0. - Peter Luschny, Jun 29 2012
E.g.f.: G(0) where G(k) = 2*k + 1 - x*(2*k+1)/(x + (2*k+2)/(1 + x/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 13 2013
a(n) = numerator(2*n*sum(k=0..2*n, (2*n+k-2)! *sum(j=1..k, ((-1)^(j+1) * stirling1(2*n+j,j)) / ((k-j)!*(2*n+j)!)))), n>0. [Vladimir Kruchinin, Mar 15 2013]
E.g.f.: E(0) where E(k) = 2*k+1 - x/(2 + x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 16 2013
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EXAMPLE
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B_{2n} = [ 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510,... ].
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MAPLE
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A000367 := n -> numer(bernoulli(2*n)):
# Illustrating an algorithmic approach:
S := proc(n, k) option remember; if k=0 then `if`(n=0, 1, 0) else S(n, k-1) + S(n-1, n-k) fi end: Bernoulli2n := n -> `if`(n = 0, 1, (-1)^n * S(2*n-1, 2*n-1)*n/(2^(2*n-1)*(1-4^n))); A000367 := n -> numer(Bernoulli2n(n)); seq(A000367(n), n=0..20); # Peter Luschny, Jul 08 2012
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MATHEMATICA
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Numerator[ BernoulliB[ 2*Range[0, 20]]] (* Jean-François Alcover, Oct 16 2012 *)
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PROG
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(PARI) a(n)=numerator(bernfrac(2*n))
# (Python) The objective of this implementation is efficiency.
# n -> [a(0), a(1), ..., a(n)] for n > 0.
from fractions import Fraction
def A000367_list(n): # Bernoulli numerators
....T = [0 for i in range(1, n+2)]
....T[0] = 1; T[1] = 1
....for k in range(2, n+1):
........T[k] = (k-1)*T[k-1]
....for k in range(2, n+1):
........for j in range(k, n+1):
............T[j] = (j-k)*T[j-1]+(j-k+2)*T[j]
....a = 0; b = 6; s = 1
....for k in range(1, n+1):
........T[k] = s*Fraction(T[k]*k, b).numerator
........h = b; b = 20*b - 64*a; a = h; s = -s
....return T
print(A000367_list(100)) # Peter Luschny, Aug 09 2011
(Maxima)
B(n):=2*n*sum((2*n+k-2)!*sum(((-1)^(j+1)*stirling1(2*n+j, j))/((k-j)!*(2*n+j)!), j, 1, k), k, 0, 2*n);
makelist(num(B(n)), n, 0, 10); [Vladimir Kruchinin, Mar 15 2013]
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CROSSREFS
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B_n gives A027641/A027642. See A027641 for full list of references, links, formulae, etc.
See A002445 for denominators.
Cf. also A002882, A003245, A127187, A127188.
Sequence in context: A090947 A176840 * A176546 A092133 A071772 A201005
Adjacent sequences: A000364 A000365 A000366 * A000368 A000369 A000370
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KEYWORD
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sign,frac,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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