A027642 - OEIS

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

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.

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

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

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, ...

(-1)^n*sum( (-1)^'m'*'m'!*stirling2(n, 'm')/('m'+1), 'm'=0..n);

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 *)

(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)

(Sage)

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

(Python)

from sympy import bernoulli

[bernoulli(i).denominator() for i in range(51)] # Indranil Ghosh, Mar 18 2017

See A027641 (numerators) for full list of references, links, formulas, etc.