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A002790
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Denominators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).
(Formerly M1559 N0608)
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23
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1, 2, 6, 4, 30, 12, 84, 24, 90, 20, 132, 24, 5460, 840, 360, 16, 1530, 180, 7980, 840, 13860, 440, 1656, 720, 81900, 6552, 216, 112, 3480, 240, 114576, 7392, 117810, 2380, 1260, 72, 3838380, 207480, 32760, 560, 568260, 27720, 238392, 55440, 869400, 2576, 236880
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OFFSET
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0,2
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COMMENTS
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The numerators are given in A002657.
These coefficients (with alternating signs) are also known as the Nørlund [or Norlund, Noerlund or Nörlund] numbers.
A simple series with the signless Cauchy numbers of second type C2(n) leads to Euler's constant: gamma = 1 - Sum_{n >=1} C2(n)/(n*(n+1)!) = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see references [Blagouchine] below, as well as A075266 and A262235. - Iaroslav V. Blagouchine, Sep 15 2015
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REFERENCES
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L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 136.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..1000
Ibrahim M. Alabdulmohsin, The Language of Finite Differences, in Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Springer, Cham, pp 133-149.
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.
Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.
C. H. Karlson & N. J. A. Sloane, Correspondence, 1974
Guodong Liu, Some computational formulas for Norlund numbers, Fib. Quart., 45 (2007), 133-137.
Guo-Dong Liu, H. M. Srivastava, 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.
Rui-Li Liu and Feng-Zhen Zhao, Log-concavity of two sequences related to Cauchy numbers of two kinds, Online Journal of Analytic Combinatorics, Issue 14 (2019), #09.
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math. 306 (2006), no. 16, 1906-1920.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951. [Annotated scan of pages 135, 136 only]
N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 461.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924; page 461 [Annotated scanned copy of pages 144-151 and 456-463]
Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., 309 (2009), 3830-3842.
Index entries for sequences related to Bernoulli numbers.
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FORMULA
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Denominator of integral of x(x+1)...(x+n-1) from 0 to 1.
E.g.f.: -x/((1-x)*log(1-x)). - Corrected by Iaroslav V. Blagouchine, May 07 2016.
Denominator of Sum_{k=0..n} (-1)^k A008275(n,k)/(k+1). - Peter Luschny, Apr 28 2009
a(n) = A091137(n)/n!. - Paul Curtz, Nov 27 2008
a(n) = denominator(n!*v(n)), where v(n) = 1 - Sum_{i=0..n-1} v(i)/(n-i+1), v(0)=1. - Vladimir Kruchinin, Aug 28 2013
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EXAMPLE
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1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ...
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MAPLE
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A002790 := proc(n)
denom(add((-1)^k*stirling1(n, k)/(k+1), k=0..n)) ;
end proc: # Peter Luschny, Apr 28 2009
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MATHEMATICA
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Table[ Denominator[ NorlundB[n, n]], {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Dec 30 2010 *)
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PROG
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(Maxima)
v(n):=if n=0 then 1 else 1-sum(v(i)/(n-i+1), i, 0, n-1);
makelist(denom(n!*v(n)), n, 0, 10); /* Vladimir Kruchinin, Aug 28 2013 */
(MAGMA) m:=60; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-x/((1-x)*Log(1-x)) )); [Denominator(Factorial(n-1)*b[n]): n in [1..m-1]]; // G. C. Greubel, Oct 28 2018
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CROSSREFS
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Cf. A002657, A075266, A075267, A262235.
See also A002208, A002209, A002206, A002207, A006232, A006233.
Sequence in context: A324922 A329886 A064538 * A108951 A181822 A346107
Adjacent sequences: A002787 A002788 A002789 * A002791 A002792 A002793
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KEYWORD
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nonn,frac,nice,easy
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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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