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A027642
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Denominator of Bernoulli number B_n.
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137
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1, 2, 6, 1, 30, 1, 42, 1, 30, 1, 66, 1, 2730, 1, 6, 1, 510, 1, 798, 1, 330, 1, 138, 1, 2730, 1, 6, 1, 870, 1, 14322, 1, 510, 1, 6, 1, 1919190, 1, 6, 1, 13530, 1, 1806, 1, 690, 1, 282, 1, 46410, 1, 66, 1, 1590, 1, 798, 1, 870, 1, 354, 1, 56786730, 1
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OFFSET
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0,2
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COMMENTS
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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. [From Gary W. Adamson, Aug 09 2008]
Julius Worpitzky's 1883 algorithm for generating Bernoulli numbers is described in A028246 [From 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 map 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.) [From Peter Luschny, Apr 29 2009]
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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.
Clausen, Thomas, "Lehrsatz aus einer Abhandlung Ueber die Bernoullischen Zahlen", Astr. Nachr. 17 (1840), 351-352. [From Peter Luschny, Apr 29 2009]
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.
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.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.
Wikipedia (Bernoulli numbers) [From Gary W. Adamson, Aug 09 2008]
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LINKS
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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].
K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), #00.2.9.
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Hisanori Mishima, Factorizations of many number sequences
Index entries for sequences related to Bernoulli numbers.
Index entries for "core" sequences
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FORMULA
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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.: 2*(x-1)/(x*Q(0) - 2) where Q(k) = 1 + 2*x*(k+1)/((2*k+1)*(2*k+3) - x*(2*k+1)*(2*k+3)^2/(x*(2*k+3) + 4*(k+1)*(k+2)/Q(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 26 2013
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
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EXAMPLE
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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, ...
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MAPLE
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(-1)^n*sum( (-1)^'m'*'m'!*stirling2(n, 'm')/('m'+1), 'm'=0..n);
A027642 := proc(n) denom(bernoulli(n)) ; end: # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 08 2009]
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MATHEMATICA
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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[n_] := Times @@ Select[Divisors[n] + 1, PrimeQ]; Table[a[n], {n, 0, 100}] (* From Jean-François Alcover, Mar 12 2012, after Ilan Vardi, when direct computation for large n is unfeasible *)
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PROG
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(PARI) a(n)=if(n<0, 0, denominator(bernfrac(n)))
(MAGMA) [Denominator(Bernoulli(n)): n in [0..150]]; // Vincenzo Librandi, Mar 29 2011
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CROSSREFS
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See A027641 (numerators) for full list of references, links, formulae, etc.
Cf. A002882, A003245, A127187, A127188, A138243, A028246, A143343, A080092, A141056, A027760.
Sequence in context: A221913 A180512 A132181 * A117214 A185972 A182918
Adjacent sequences: A027639 A027640 A027641 * A027643 A027644 A027645
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KEYWORD
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nonn,frac,easy,core,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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