A002790 Denominators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}). (Formerly M1559 N0608)

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

Offset

0,2

Comments

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

References

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).

Links

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.

Formula

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

Example

1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ...

Maple

A002790 := proc(n)
    denom(add((-1)^k*stirling1(n, k)/(k+1), k=0..n)) ;
end proc: # Peter Luschny, Apr 28 2009

Mathematica

Table[ Denominator[ NorlundB[n, n]], {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Dec 30 2010 *)

Prog

(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

Crossrefs

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

Keyword

nonn,frac,nice,easy

Author

N. J. A. Sloane

Status

approved