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

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

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• nice
• easy
• changed

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

a(n) appears to be divisible by n+1. - Hal M. Switkay, Aug 15 2025

Examples

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

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.

Crossrefs

Cf. A002657, A075266, A075267, A262235.

See also A002208, A002209, A002206, A002207, A006232, A006233.

Programs

Magma

m:=60; R:=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

Maple

A002790 := proc(n)
    denom(add((-1)^k*Stirling1(n, k)/(k+1), k=0..n)) ;
end proc: # Peter Luschny, Apr 28 2009
v := proc(n) option remember; ifelse(n=0, 1, 1 - add(v(i)/(n-i+1), i=0..n-1)) end:
seq(denom(n!*v(n)), n = 0..46); # after Vladimir Kruchinin, Peter Luschny, Aug 17 2025

Mathematica

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

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 */

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) = 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