OFFSET
0,2
COMMENTS
Row products of A138243. - Mats Granvik, Mar 08 2008
Equals row products of triangle A143343 and for a(n) > 1, row products of triangle A080092. - Gary W. Adamson, Aug 09 2008
Julius Worpitzky's 1883 algorithm for generating Bernoulli numbers is described in A028246. - Gary W. Adamson, Aug 09 2008
The sequence of denominators of B_n is defined here by convention, not by necessity. The convention amounts to mapping 0 to the rational number 0/1. It might be more appropriate to regard numerators and denominators of the Bernoulli numbers as independent sequences N_n and D_n which combine to B_n = N_n / D_n. This is suggested by the theorem of Clausen which describes the denominators as the sequence D_n = 1, 2, 6, 2, 30, 2, 42, ... which combines with N_n = 1, -1, 1, 0, -1, 0, ... to the sequence of Bernoulli numbers. (Cf. A141056 and A027760.) - Peter Luschny, Apr 29 2009
REFERENCES
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.
Jacob Bernoulli, Ars Conjectandi, Basel: Thurneysen Brothers, 1713. See page 97.
Thomas Clausen, "Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen", Astr. Nachr. 17 (1840), 351-352 (see P. Luschny link).
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.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.
Roger Plymen, The Great Prime Number Race, AMS, 2020. See pp. 8-10.
LINKS
T. D. Noe, Table of n, a(n) for n = 0..10000
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].
Beáta Bényi and Péter Hajnal, Poly-Bernoulli Numbers and Eulerian Numbers, arXiv:1804.01868 [math.CO], 2018.
K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
H. W. Gould and J. Quaintance, Bernoulli Numbers and a New Binomial Transform Identity, J. Int. Seq. 17 (2014) # 14.2.2
A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq. 17 (2014) # 14.4.6
Peter Luschny, Generalized Clausen numbers: definition and application.
R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
A. F. Neto, Carlitz's Identity for the Bernoulli Numbers and Zeon Algebra, J. Int. Seq. 18 (2015) # 15.5.6.
Carl Pomerance and Samuel S. Wagstaff Jr, The denominators of the Bernoulli numbers, arXiv:2105.13252 [math.NT], 2021.
J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 5.
Matthew Roughan, The Polylogarithm Function in Julia, arXiv:2010.09860 [math.NA], 2020.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
Wikipedia, Bernoulli number
FORMULA
E.g.f: x/(exp(x) - 1); take denominators.
Let E(x) be the e.g.f., then E(x) = U(0), where U(k) = 2*k + 1 - x*(2*k+1)/(x + (2*k+2)/(1 + x/U(k+1))); (continued fraction, 3-step). - Sergei N. Gladkovskii, Jun 25 2012
E.g.f.: x/(exp(x)-1) = E(0) where E(k) = 2*k+1 - x/(2 + x/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Mar 16 2013
E.g.f.: x/(exp(x)-1) = 2*E(0) - 2*x, where E(k)= x + (k+1)/(1 + 1/(1 - x/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 10 2013
E.g.f.: x/(exp(x)-1) = (1-x)/E(0), where E(k) = 1 - x*(k+1)/(x*(k+1) + (k+2-x)*(k+1-x)/E(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 21 2013
E.g.f.: conjecture: x/(exp(x)-1) = T(0)/2 - x, where T(k) = 8*k+2 + x/( 1 - x/( 8*k+6 + x/( 1 - x/T(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Nov 24 2013
EXAMPLE
B_n sequence begins 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, ...
MAPLE
(-1)^n*sum( (-1)^'m'*'m'!*stirling2(n, 'm')/('m'+1), 'm'=0..n);
A027642 := proc(n) denom(bernoulli(n)) ; end: # Zerinvary Lajos, Apr 08 2009
MATHEMATICA
Table[ Denominator[ BernoulliB[n]], {n, 0, 68}] (* Robert G. Wilson v, Oct 11 2004 *)
Denominator[ Range[0, 68]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 68}], x]]
(* Alternative code using Clausen Theorem: *)
A027642[k_Integer]:=If[EvenQ[k], Times@@Table[Max[1, Prime[i]*Boole[Divisible[k, Prime[i]-1]]], {i, 1, PrimePi[2k]}], 1+KroneckerDelta[k, 1]]; (* Enrique Pérez Herrero, Jul 15 2010 *)
a[0] = 1; a[1] = 2; a[n_?OddQ] = 1; a[n_] := Times @@ Select[Divisors[n] + 1, PrimeQ]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Mar 12 2012, after Ilan Vardi, when direct computation for large n is unfeasible *)
PROG
(PARI) a(n)=if(n<0, 0, denominator(bernfrac(n)))
(PARI) a(n) = if(n == 0 || (n > 1 && n % 2), 1, vecprod(select(x -> isprime(x), apply(x -> x + 1, divisors(n))))); \\ Amiram Eldar, Apr 24 2024
(Magma) [Denominator(Bernoulli(n)): n in [0..150]]; // Vincenzo Librandi, Mar 29 2011
(Haskell)
a027642 n = a027642_list !! n
a027642_list = 1 : map (denominator . sum) (zipWith (zipWith (%))
(zipWith (map . (*)) (tail a000142_list) a242179_tabf) a106831_tabf)
-- Reinhard Zumkeller, Jul 04 2014
(Sage)
def A027642_list(len):
f, R, C = 1, [1], [1]+[0]*(len-1)
for n in (1..len-1):
f *= n
for k in range(n, 0, -1):
C[k] = C[k-1] / (k+1)
C[0] = -sum(C[k] for k in (1..n))
R.append((C[0]*f).denominator())
return R
A027642_list(62) # Peter Luschny, Feb 20 2016
(Python)
from sympy import bernoulli
[bernoulli(i).denominator() for i in range(51)] # Indranil Ghosh, Mar 18 2017
CROSSREFS
See A027641 (numerators) for full list of references, links, formulas, etc.
KEYWORD
nonn,frac,easy,core,nice
AUTHOR
STATUS
approved