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%I M3790 N1545 #129 Apr 15 2022 13:04:17
%S 1,1,5,9,251,475,19087,36799,1070017,2082753,134211265,262747265,
%T 703604254357,1382741929621,8164168737599,5362709743125,
%U 8092989203533249,15980174332775873,12600467236042756559
%N Numerators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).
%C 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]
%C 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]
%C 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
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
%D P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
%D Louis Melville Milne-Thompson, Calculus of Finite Differences, 1951, p. 136.
%D N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002657/b002657.txt">Table of n, a(n) for n = 0..100</a>
%H Ibrahim M. Alabdulmohsin, <a href="https://doi.org/10.1007/978-3-319-74648-7_7">The Language of Finite Differences</a>, in Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Springer, Cham, 2018, pp. 133-149.
%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jnt.2015.06.012">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.</a> Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. <a href="http://arxiv.org/abs/1501.00740">arXiv version</a>, arXiv:1501.00740 [math.NT], 2015.
%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jmaa.2016.04.032">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</a>, Journal of Mathematical Analysis and Applications (Elsevier), 2016. <a href="http://arxiv.org/abs/1408.3902">arXiv version</a>, arXiv:1408.3902 [math.NT], 2014-2016.
%H Iaroslav V. Blagouchine, <a href="http://math.colgate.edu/~integers/sjs3/sjs3.Abstract.html">Three notes on Ser's and Hasse's representation for the zeta-functions</a>, Integers (2018) 18A, Article #A3.
%H Takao Komatsu, <a href="http://doi.org/10.2206/kyushujm.69.125">Convolution Identities for Cauchy Numbers of the Second Kind</a>, Kyushu Journal of Mathematics, Vol. 69, No. 1 (2015), pp. 125-144.
%H Guodong Liu, <a href="http://www.fq.math.ca/Papers1/45-2/quartliu02_2007.pdf">Some computational formulas for Norlund numbers</a>, Fib. Quart., Vol. 45, No. 2 (2007), pp. 133-137.
%H Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Srivastava/sriva3.html">Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers</a>, J. Int. Seq., Vol. 17 (2014), Article 14.4.6.
%H Rui-Li Liu and Feng-Zhen Zhao, <a href="https://hosted.math.rochester.edu/ojac/vol14/183.pdf">Log-concavity of two sequences related to Cauchy numbers of two kinds</a>, Online Journal of Analytic Combinatorics, Issue 14 (2019), #09.
%H Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, <a href="http://dx.doi.org/10.1016/j.disc.2006.03.065">The Cauchy numbers</a>, Discrete Math., Vol. 306, No. 16 (2006), pp. 1906-1920.
%H Louis Melville Milne-Thompson, <a href="/A002657/a002657.pdf"> Calculus of Finite Differences</a>, 1951. [Annotated scan of pages 135, 136 only]
%H N. E. Nørlund, <a href="http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN373206070">Vorlesungen ueber Differenzenrechnung</a> Springer, 1924, p. 461.
%H N. E. Nörlund, <a href="/A001896/a001896_1.pdf">Vorlesungen über Differenzenrechnung</a>, Springer-Verlag, Berlin, 1924; page 461 [Annotated scanned copy of pages 144-151 and 456-463]
%H Michael O. Rubinstein, <a href="https://doi.org/10.1007/s11139-010-9276-8">Identities for the Riemann zeta function</a>, Ramanujan J., Vol. 27, No. 1 (2012), pp. 29-42; <a href="https://arxiv.org/abs/0812.2592">arXiv preprint</a>, arXiv:0812.2592 [math.NT], 2008-2009.
%H Feng-Zhen Zhao, <a href="http://dx.doi.org/10.1016/j.disc.2008.10.013">Sums of products of Cauchy numbers</a>, Discrete Math., Vol. 309, No. 12 (2009), pp. 3830-3842.
%H <a href="/index/Be#Bernoulli">Index entries for sequences related to Bernoulli numbers.</a>
%F Numerator of integral of x(x+1)...(x+n-1) from 0 to 1.
%F 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
%F Numerator of Sum_{k=0..n} (-1)^(n-k) A008275(n,k)/(k+1). - _Peter Luschny_, Apr 28 2009
%F 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
%e 1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ...
%p seq(numer(add((-1)^(n-k)*Stirling1(n,k)/(k+1),k=0..n)),n=0..10); # _Peter Luschny_, Apr 28 2009
%t Table[Abs[Numerator[NorlundB[n,n]]],{n,0,30}](* _Vladimir Joseph Stephan Orlovsky_, Dec 30 2010 *)
%t a[ n_] := If[ n < 0, 0, (-1)^n Numerator @ NorlundB[ n, n]]; (* _Michael Somos_, Jul 12 2014 *)
%t a[ n_] := If[ n < 0, 0, Numerator@Integrate[ Pochhammer[ x, n], {x, 0, 1}]]; (* _Michael Somos_, Jul 12 2014 *)
%t a[ n_] := If[ n < 0, 0, Numerator[ n! SeriesCoefficient[ -x / ((1 - x) Log[ 1 - x]), {x, 0, n}]]]; (* _Michael Somos_, Jul 12 2014 *)
%t a[ n_] := If[ n < 0, 0, (-1)^n Numerator[ n! SeriesCoefficient[ (x / (Exp[x] - 1))^n, {x, 0, n}]]]; (* _Michael Somos_, Jul 12 2014 *)
%o (Maxima) v(n):=if n=0 then 1 else 1-sum(v(i)/(n-i+1),i,0,n-1);
%o makelist(num(n!*v(n)),n,0,10); /* _Vladimir Kruchinin_, Aug 28 2013 */
%o 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
%Y Cf. A002206, A002207, A002208, A002209, A002790, A006232, A006233, A075266, A075267, A262235.
%K nonn,frac,easy,nice
%O 0,3
%A _N. J. A. Sloane_
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