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A006233 Denominators of Cauchy numbers of first type.

%I A006233 M1558 #87 Feb 16 2025 08:32:29
%S A006233 1,2,6,4,30,4,84,24,90,20,132,8,5460,840,360,48,1530,4,1596,168,1980,
%T A006233 1320,8280,80,81900,6552,1512,112,3480,80,114576,7392,117810,7140,
%U A006233 1260,8,3838380,5928,936,48,81180,440,1191960,55440,869400,38640,236880,224
%N A006233 Denominators of Cauchy numbers of first type.
%C A006233 The corresponding numerators are given in A006232.
%C A006233 The signed rationals A006232(n)/a(n) provide the a-sequence for the Stirling2 Sheffer matrix A048993. See the W. Lang link concerning Sheffer a- and z-sequences.
%C A006233 Cauchy numbers of the first type are also called Bernoulli numbers of the second kind.
%D A006233 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
%D A006233 H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 1946, p. 259.
%D A006233 L. Jolley, Summation of Series, Chapman and Hall, London, 1925, pp. 14-15 (formula 70).
%D A006233 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006233 T. D. Noe, <a href="/A006233/b006233.txt">Table of n, a(n) for n = 0..1000</a>
%H A006233 A. Adelberg, <a href="http://dx.doi.org/10.1006/jnth.1998.2278">2-adic congruences of Norland numbers and of Bernoulli numbers of the second kind</a>, J. Number Theory, 73 (1998), 47-58.
%H A006233 I. S. Gradsteyna and I. M. Ryzhik, <a href="http://mathtable.com/gr/index.html">Table of integrals, series and products</a>, (1980), page 2 (formula 0.131).
%H A006233 Wolfdieter Lang, <a href="/A006233/a006233.txt">Sheffer a- and z-sequences</a>
%H A006233 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 A006233 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. 306 (2006), no. 16, 1906-1920.
%H A006233 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BernoulliNumberoftheSecondKind.html">Bernoulli Number of the Second Kind</a>
%H A006233 Ming Wu and Hao Pan, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Papers1/45-2/quartpan02_2007.pdf">Sums of products of Bernoulli numbers of the second kind</a>, Fib. Quart., 45 (2007), 146-150.
%H A006233 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., 309 (2009), 3830-3842.
%F A006233 Denominator of integral of x(x-1)...(x-n+1) from 0 to 1.
%F A006233 E.g.f.: x/log(1+x).
%F A006233 Denominator of Sum_{k=0..n} A048994(n,k)/(k+1). [_Peter Luschny_, Apr 28 2009]
%F A006233 a(n) = denominator(f(n) * n!), where f(0) = 1, f(n) = Sum_{k=0..n-1} (-1)^(n-k+1) * f(k) / (n-k+1). - _Daniel Suteu_, Feb 23 2018
%F A006233 Sum_{k = 1..n} (1/k) = A001620 + log(n) + 1/(2*n) - Sum_{k >= 2} abs((A006232(k)/a(k)/k/(Product_{j = 0..k-1} (n-j)))), (see I. S. Gradsteyn, I. M. Ryzhik). - _A.H.M. Smeets_, Nov 14 2018
%e A006233 1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90,...
%p A006233 seq(denom(add(stirling1(n,k)/(k+1),k=0..n)),n=0..12); # _Peter Luschny_, Apr 28 2009
%t A006233 With[{nn=50},Denominator[CoefficientList[Series[x/Log[1+x],{x,0,nn}],x] Range[0,nn]!]] (* _Harvey P. Dale_, Oct 28 2011 *)
%t A006233 a[n_] := Sum[ StirlingS1[n, k]/(k+1), {k, 0, n}] // Denominator; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Jan 10 2013, after _Peter Luschny_ *)
%t A006233 Join[{1}, Array[Abs@Denominator[ Integrate[Product[(x - k), {k, 0, # - 1}], {x, 0, 1}]] &, 50]] (* _Michael De Vlieger_, Nov 13 2018 *)
%o A006233 (PARI) for(n=0,50, print1(denominator( sum(k=0,n, stirling(n, k, 1)/(k+1)) ), ", ")) \\ _G. C. Greubel_, Nov 13 2018
%o A006233 (Magma) [Denominator((&+[StirlingFirst(n,k)/(k+1): k in [0..n]])): n in [0..50]]; // _G. C. Greubel_, Nov 13 2018
%o A006233 (Sage)
%o A006233 def A006233_list(len):
%o A006233     f, R, C = 1, [1], [1]+[0]*(len-1)
%o A006233     for n in (1..len-1):
%o A006233         for k in range(n, 0, -1):
%o A006233             C[k] = -C[k-1] * k / (k + 1)
%o A006233         C[0] = -sum(C[k] for k in (1..n))
%o A006233         R.append((C[0]*f).denominator())
%o A006233         f *= n+1
%o A006233     return R
%o A006233 print(A006233_list(50))  # _G. C. Greubel_, Nov 13 2018
%o A006233 (Python) # Results are abs values
%o A006233 from fractions import gcd
%o A006233 aa, n, sden = [0, 1], 1, 1
%o A006233 print(0, 1)
%o A006233 while n < 20:
%o A006233     j, snom, sden, a = 1, 0, (n+1)*sden, 0
%o A006233     while j < len(aa):
%o A006233         snom, j = snom+aa[j]*(sden//(j+1)), j+1
%o A006233     nom, den = snom, sden
%o A006233     print(n,den//gcd(nom,den))
%o A006233     aa, j = aa+[-aa[j-1]], j-1
%o A006233     while j > 0:
%o A006233         aa[j], j = n*aa[j]-aa[j-1], j-1
%o A006233     n += 1 # _A.H.M. Smeets_, Nov 14 2018
%o A006233 (Python)
%o A006233 from fractions import Fraction
%o A006233 from sympy.functions.combinatorial.numbers import stirling
%o A006233 def A006233(n): return sum(Fraction(stirling(n,k,kind=1,signed=True),k+1) for k in range(n+1)).denominator # _Chai Wah Wu_, Jul 09 2023
%Y A006233 Cf. A006232 (numerators), A002206, A002207, A002208, A002209, A002657, A002790.
%K A006233 nonn,frac,nice,easy
%O A006233 0,2
%A A006233 _N. J. A. Sloane_