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A000367 Numerators of Bernoulli numbers B_2n.
(Formerly M4039 N1677)

%I M4039 N1677

%S 1,1,-1,1,-1,5,-691,7,-3617,43867,-174611,854513,-236364091,8553103,

%T -23749461029,8615841276005,-7709321041217,2577687858367,

%U -26315271553053477373,2929993913841559,-261082718496449122051

%N Numerators of Bernoulli numbers B_2n.

%D 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.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.

%D 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.

%D G. Everest, A. van der Poorten, I. Shparlinski and T. Ward, Recurrence Sequences, Amer. Math. Soc., 2003; see esp. p. 255.

%D H. H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.

%D F. Lemmermeyer, Reciprocity Laws From Euler to Eisenstein, Springer-Verlag, 2000, p. 330.

%D H. Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H Simon Plouffe, <a href="/A000367/b000367.txt">Table of n, a(n) for n = 0..249</a> [taken from link below]

%H M. Abramowitz and I. A. Stegun, eds., <a href="http://apps.nrbook.com/abramowitz_and_stegun/index.html">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

%H J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles</a>, arXiv:math/0109108 [math.NT], 2001.

%H Richard P. Brent and David Harvey, <a href="http://arxiv.org/abs/1108.0286">Fast computation of Bernoulli, Tangent and Secant numbers</a>, arXiv preprint arXiv:1108.0286, 2011

%H J. Butcher, <a href="http://www.math.auckland.ac.nz/~butcher/miniature/miniature18.pdf">Some applications of Bernoulli numbers</a>

%H C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php/BernoulliNumber.html">Bernoulli number</a>

%H F. N. Castro, O. E. González, L. A. Medina, <a href="http://emmy.uprrp.edu/lmedina/papers/eulerian/EulerianFinal.pdf">The p-adic valuation of Eulerian numbers: trees and Bernoulli numbers</a>, 2014.

%H R. Jovanovic, <a href="http://milan.milanovic.org/math/english/bernoulli/bernoulli.html">Bernoulli numbers and the Pascal triangle</a>

%H M. Kaneko, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL3/KANEKO/AT-kaneko.html">The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer Sequences, 3 (2000), #00.2.9.

%H B. C. Kellner, <a href="http://arXiv.org/abs/math.NT/0409223">On irregular prime power divisors of the Bernoulli numbers</a>, arXiv:math/0409223 [math.NT], 2004-2005.

%H B. C. Kellner, <a href="http://arxiv.org/abs/math/0411498">The structure of Bernoulli numbers</a>, arXiv:math/0411498 [math.NT], 2004.

%H C. Lin and L. Zhipeng, <a href="http://arXiv.org/abs/math.HO/0408082">On Bernoulli numbers and its properties</a>, arXiv:math/0408082 [math.HO], 2004.

%H Guo-Dong Liu, H. M. Srivastava, Hai-Quing Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Srivastava/sriva3.html">Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers</a>, J. Int. Seq. 17 (2014) # 14.4.6

%H H.-M. Liu, S-H. Qi, S.-Y. Ding, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Liu/liu4.html">Some Recurrence Relations for Cauchy Numbers of the First Kind</a>, JIS 13 (2010) # 10.3.8.

%H S. O. S. Math, <a href="http://www.sosmath.com/tables/bernoulli/bernoulli.html">Bernoulli and Euler Numbers</a>

%H R. Mestrovic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.html">On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers</a>, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.

%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha134.htm">Factorizations of many number sequences</a>

%H Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/matha1/matha1341.htm">Factorizations of many number sequences</a>

%H Niels Nielsen, <a href="http://gallica.bnf.fr/ark:/12148/bpt6k62119c">Traite Elementaire des Nombres de Bernoulli</a>, Gauthier-Villars, 1923, pp. 398.

%H N. E. Nörlund, <a href="/A001896/a001896_1.pdf">Vorlesungen über Differenzenrechnung</a>, Springer-Verlag, Berlin, 1924 [Annotated scanned copy of pages 144-151 and 456-463]

%H Simon Plouffe, <a href="http://www.lacim.uqam.ca/~plouffe/ber250000.txt">The 250,000th Bernoulli Number</a>

%H Simon Plouffe, <a href="http://www.ibiblio.org/gutenberg/etext01/brnll10.txt">The First 498 Bernoulli numbers</a> [Project Gutenberg Etext]

%H S. Ramanujan, <a href="http://www.imsc.res.in/~rao/ramanujan/collectedpapers/Bernoulli/bernoulli1.html">Some Properties of Bernoulli's Numbers</a>

%H Zhi-Hong Sun, <a href="http://dx.doi.org/10.1016/j.disc.2007.03.038">Congruences involving Bernoulli polynomials</a>, Discr. Math., 308 (2007), 71-112.

%H S. S. Wagstaff, <a href="http://www.cerias.purdue.edu/homes/ssw/bernoulli/bnum">Prime factors of the absolute values of Bernoulli numerators</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BernoulliNumber.html">Bernoulli Number.</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>

%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>

%F E.g.f: x/(exp(x) - 1); take numerators of even powers.

%F 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).

%F If n>=3 is prime, then 12*|a((n+1)/2)|==(-1)^((n-1)/2)*A002445((n+1)/2) (mod n). - _Vladimir Shevelev_, Sep 04 2010

%F a(n) = numerator(-I*(2*n)!/(Pi*(1-2*n))*integral(log(1-1/t)^(1-2*n) dt, t=0..1)). - _Gerry Martens_, May 17 2011, corrected by _Vaclav Kotesovec_, Oct 22 2014

%F a(n) = numerator((-1)^(n+1)*(2*Pi)^(-2*n)*(2*n)!*Li_{2*n}(1)) for n > 0. - _Peter Luschny_, Jun 29 2012

%F 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

%F 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

%F 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

%F E.g.f.: E(0)- x, where E(k)= x+k+1 - x*(k+1)/E(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Jul 14 2013

%e B_{2n} = [ 1, 1/6, -1/30, 1/42, -1/30, 5/66, -691/2730, 7/6, -3617/510,... ].

%p A000367 := n -> numer(bernoulli(2*n)):

%p # Illustrating an algorithmic approach:

%p 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

%t Numerator[ BernoulliB[ 2*Range[0, 20]]] (* _Jean-François Alcover_, Oct 16 2012 *)

%o (PARI) a(n)=numerator(bernfrac(2*n))

%o (Python) # The objective of this implementation is efficiency.

%o # n -> [a(0), a(1), ..., a(n)] for n > 0.

%o from fractions import Fraction

%o def A000367_list(n): # Bernoulli numerators

%o ....T = [0 for i in range(1, n+2)]

%o ....T[0] = 1; T[1] = 1

%o ....for k in range(2, n+1):

%o ........T[k] = (k-1)*T[k-1]

%o ....for k in range(2, n+1):

%o ........for j in range(k, n+1):

%o ............T[j] = (j-k)*T[j-1]+(j-k+2)*T[j]

%o ....a = 0; b = 6; s = 1

%o ....for k in range(1, n+1):

%o ........T[k] = s*Fraction(T[k]*k, b).numerator

%o ........h = b; b = 20*b - 64*a; a = h; s = -s

%o ....return T

%o print(A000367_list(100)) # _Peter Luschny_, Aug 09 2011

%o (Maxima)

%o B(n):=if n=0 then 1 else 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);

%o makelist(num(B(n)),n,0,10); /* _Vladimir Kruchinin_, Mar 15 2013, fixed by _Vaclav Kotesovec_, Oct 22 2014 */

%Y B_n gives A027641/A027642. See A027641 for full list of references, links, formulas, etc.

%Y See A002445 for denominators.

%Y Cf. also A002882, A003245, A127187, A127188.

%K sign,frac,nice,changed

%O 0,6

%A _N. J. A. Sloane_

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Last modified February 17 23:47 EST 2018. Contains 299297 sequences. (Running on oeis4.)