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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.
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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