Search: id:A027642
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%I A027642
%S A027642 1,2,6,1,30,1,42,1,30,1,66,1,2730,1,6,1,510,1,798,1,330,1,138,1,
%T A027642 2730,1,6,1,870,1,14322,1,510,1,6,1,1919190,1,6,1,13530,1,1806,
%U A027642 1,690,1,282,1,46410,1,66,1,1590,1,798,1,870,1,354,1,56786730,1
%N A027642 Denominator of Bernoulli number B_n.
%C A027642 Row products of A138243. - Mats O. Granvik (mgranvik(AT)abo.fi), Mar 
               08 2008
%C A027642 Equals row products of triangle A143343 and for a(n)>1, row products 
               of triangle A080092. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 09 2008]
%C A027642 Julius Worpitzky's 1883 algorithm for generating Bernoulli numbers is 
               described in A028246 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 09 2008]
%C A027642 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 (peter(AT)luschny.de), Apr 29 2009]
%D A027642 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 A027642 Clausen, Thomas, "Lehrsatz aus einer Abhandlung Ueber die Bernoullischen 
               Zahlen", Astr. Nachr. 17 (1840), 351-352. [From Peter Luschny (peter(AT)luschny.de), 
               Apr 29 2009]
%D A027642 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
%D A027642 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 A027642 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.
%D A027642 L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.
%D A027642 Wikipedia (Bernoulli numbers) [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 09 2008]
%H A027642 T. D. Noe, <a href="b027642.txt">Table of n, a(n) for n = 0..10000</a>
%H A027642 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A027642 K.-W. Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Algorithms for Bernoulli numbers and Euler numbers</a>, J. Integer 
               Sequences, 4 (2001), #01.1.6.
%H A027642 M. Kaneko, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Akiyama-Tanigawa algorithm for Bernoulli numbers</a>, J. Integer 
               Sequences, 3 (2000), #00.2.9.
%H A027642 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha103.htm">Factorizations of many number sequences</a>
%H A027642 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha134.htm">Factorizations of many number sequences</a>
%H A027642 Hisanori Mishima, <a href="http://www.asahi-net.or.jp/~KC2H-MSM/mathland/
               matha1/matha1341.htm">Factorizations of many number sequences</a>
%H A027642 <a href="Sindx_Be.html#Bernoulli">Index entries for sequences related 
               to Bernoulli numbers.</a>
%H A027642 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A027642 E.g.f: x/(e^x - 1).
%e A027642 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, ...
%p A027642 (-1)^n*sum( (-1)^'m'*'m'!*stirling2(n,'m')/('m'+1),'m'=0..n);
%p A027642 with(numtheory):seq(denom(1-bernoulli(n+2)) ,n=-2..59);# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Apr 08 2009]
%p A027642 with(numtheory):seq(denom(bernoulli(n)) ,n=0..59);# [From Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Apr 08 2009]
%t A027642 Table[ Denominator[ BernoulliB[n]], {n, 0, 68}] (from Robert G. Wilson 
               v Oct 11 2004)
%t A027642 Denominator[ Range[0, 68]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 
               68}], x]]
%o A027642 (PARI) a(n)=if(n<0, 0, denominator(bernfrac(n)))
%Y A027642 See A027641 for full list of references, links, formulae, etc.
%Y A027642 Cf. also A002882, A003245, A127187, A127188.
%Y A027642 Cf. A138243.
%Y A027642 Cf. A028246, A143343, A080092 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 
               Aug 09 2008]
%Y A027642 Cf. A141056, A027760. [From Peter Luschny (peter(AT)luschny.de), Apr 
               29 2009]
%Y A027642 Sequence in context: A111519 A008855 A132181 this_sequence A117214 A134301 
               A004544
%Y A027642 Adjacent sequences: A027639 A027640 A027641 this_sequence A027643 A027644 
               A027645
%K A027642 nonn,frac,easy,core,nice
%O A027642 0,2
%A A027642 N. J. A. Sloane (njas(AT)research.att.com).

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