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A002657
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Numerators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).
(Formerly M3790 N1545)
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27
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1, 1, 5, 9, 251, 475, 19087, 36799, 1070017, 2082753, 134211265, 262747265, 703604254357, 1382741929621, 8164168737599, 5362709743125, 8092989203533249, 15980174332775873, 12600467236042756559
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OFFSET
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0,3
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COMMENTS
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These coefficients (with alternating signs) are also known as the Nørlund [or Norlund, Noerlund or Nörlund] numbers. [After the Danish mathematician Niels Erik Nørlund (1885-1981). - Amiram Eldar, Jun 17 2021]
The denominators are found in A002790. The alternating rational sequence ((-1)^n)*a(n)/A002790(n)is the z-sequence for the Stirling2 triangle A008277(n+1,k+1), n>=k>=0. This is the Sheffer (exp(x),exp(x)-1) triangle. See the W. Lang link under A006232 for Sheffer a- and z-sequences with references, and the conversion to S. Roman's notation. The a-sequence is A006232(n)/A006233(n). - Wolfdieter Lang, Oct 06 2011 [This is the Sheffer triangle A007318*A048993. Added Jun 20 2017]
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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Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
Louis Melville 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..100
Ibrahim M. Alabdulmohsin, The Language of Finite Differences, in Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Springer, Cham, 2018, pp. 133-149.
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, 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, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.
Takao Komatsu, Convolution Identities for Cauchy Numbers of the Second Kind, Kyushu Journal of Mathematics, Vol. 69, No. 1 (2015), pp. 125-144.
Guodong Liu, Some computational formulas for Norlund numbers, Fib. Quart., Vol. 45, No. 2 (2007), pp. 133-137.
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., Vol. 17 (2014), Article 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., Vol. 306, No. 16 (2006), pp. 1906-1920.
Louis Melville 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]
Michael O. Rubinstein, Identities for the Riemann zeta function, Ramanujan J., Vol. 27, No. 1 (2012), pp. 29-42; arXiv preprint, arXiv:0812.2592 [math.NT], 2008-2009.
Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., Vol. 309, No. 12 (2009), pp. 3830-3842.
Index entries for sequences related to Bernoulli numbers.
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FORMULA
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Numerator of integral of x(x+1)...(x+n-1) from 0 to 1.
E.g.f.: -x/((1-x)*log(1-x)). (Note: the numerator of the coefficient of x^n/n! is a(n). - Michael Somos, Jul 12 2014). E.g.f. rewritten by Iaroslav V. Blagouchine, May 07 2016
Numerator of Sum_{k=0..n} (-1)^(n-k) A008275(n,k)/(k+1). - Peter Luschny, Apr 28 2009
a(n) = numerator(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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seq(numer(add((-1)^(n-k)*Stirling1(n, k)/(k+1), k=0..n)), n=0..10); # Peter Luschny, Apr 28 2009
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MATHEMATICA
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Table[Abs[Numerator[NorlundB[n, n]]], {n, 0, 30}](* Vladimir Joseph Stephan Orlovsky, Dec 30 2010 *)
a[ n_] := If[ n < 0, 0, (-1)^n Numerator @ NorlundB[ n, n]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, Numerator@Integrate[ Pochhammer[ x, n], {x, 0, 1}]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, Numerator[ n! SeriesCoefficient[ -x / ((1 - x) Log[ 1 - x]), {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, (-1)^n Numerator[ n! SeriesCoefficient[ (x / (Exp[x] - 1))^n, {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)
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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(num(n!*v(n)), n, 0, 10); /* Vladimir Kruchinin, Aug 28 2013 */
m:=25; R<x>:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-x/((1-x)*Log(1-x)) )); [Numerator(Factorial(n-1)*b[n]): n in [1..m-1]]; // G. C. Greubel, Oct 29 2018
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CROSSREFS
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Cf. A002206, A002207, A002208, A002209, A002790, A006232, A006233, A075266, A075267, A262235.
Sequence in context: A097397 A092584 A145400 * A046093 A097086 A109076
Adjacent sequences: A002654 A002655 A002656 * A002658 A002659 A002660
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KEYWORD
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nonn,frac,easy,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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