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%I M5067 #110 Apr 17 2024 11:20:10
%S 1,1,-1,1,-19,9,-863,1375,-33953,57281,-3250433,1891755,-13695779093,
%T 24466579093,-132282840127,240208245823,-111956703448001,
%U 4573423873125,-30342376302478019,56310194579604163
%N Numerators of Cauchy numbers of first type.
%C The corresponding denominators are given in A006233.
%C -a(n+1), n>=0, also numerators from e.g.f. 1/x-1/log(1+x), with denominators A075178(n). |a(n+1)|, n>=0, numerators from e.g.f. 1/x+1/log(1-x) with denominators A075178(n). For formula of unsigned a(n) see A075178.
%C The signed rationals a(n)/A006233(n) provide the a-sequence for the Stirling2 Sheffer matrix A048993. See the W. Lang link concerning Sheffer a- and z-sequences.
%C Cauchy numbers of the first type are also called Bernoulli numbers of the second kind.
%C Named after the French mathematician, engineer and physicist Augustin-Louis Cauchy (1789-1857). - _Amiram Eldar_, Jun 17 2021
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
%D Harold Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 1946, p. 259.
%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="/A006232/b006232.txt">Table of n, a(n) for n=0..100</a>
%H Arnold 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, Vol. 73, No. 1 (1998), pp. 47-58.
%H I. S. Gradsteyn 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 L. B. W. Jolley, <a href="https://archive.org/details/summationofserie00joll">Summation of Series</a>, Dover, (1961) (formula 70).
%H Wolfdieter Lang, <a href="/A006232/a006232.pdf">Sheffer a- and z-sequences</a>.
%H Wolfdieter Lang, <a href="https://arxiv.org/abs/1708.01421">On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles</a>, arXiv:1708.01421 [math.NT], August 2017.
%H Hong-Mei Liu, Shu-Hua Qi and Shu-Yan Ding, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Liu/liu4.html">Some Recurrence Relations for Cauchy Numbers of the First Kind</a>, JIS, Vol. 13 (2010), Article 10.3.8.
%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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BernoulliNumberoftheSecondKind.html">Bernoulli Numbers of the Second Kind</a>.
%H Ming Wu and Hao Pan, <a href="http://www.fq.math.ca/Papers1/45-2/quartpan02_2007.pdf">Sums of products of Bernoulli numbers of the second kind</a>, Fib. Quart., Vol. 45, No. 2 (2007), pp. 146-150.
%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.
%F Numerator of integral of x(x-1)...(x-n+1) from 0 to 1.
%F E.g.f.: x/log(1+x). (Note: the numerator of the coefficient of x^n/n! is a(n) - _Michael Somos_, Jul 12 2014)
%F Numerator of Sum_{k=0..n} A048994(n,k)/(k+1). - _Peter Luschny_, Apr 28 2009
%F Sum_{k=1..n} 1/k = C + log(n) + 1/(2n) + Sum_{k=2..inf} |a(n)|/A075178(n-1) * 1/(n*(n+1)*...*(n+k-1)) (section 0.131 in Gradshteyn and Ryzhik tables). - _Ralf Stephan_, Jul 12 2014
%F a(n) = numerator(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 Sum_{k = 1..n} (1/k) = A001620 + log(n) + 1/(2n) - Sum_{k >= 2} abs((a(k)/A006233(k)/k/(Product_{j = 0..k-1} (n-j)))), (see I. S. Gradsteyn, I. M. Ryzhik). - _A.H.M. Smeets_, Nov 14 2018
%e 1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90, ...
%p seq(numer(add(stirling1(n,k)/(k+1),k=0..n)),n=0..20); # _Peter Luschny_, Apr 28 2009
%t a[n_] := Numerator[ Sum[ StirlingS1[n, k]/(k + 1), {k, 0, n}]]; Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Nov 03 2011, after Maple *)
%t a[n_] := Numerator[ Integrate[ Gamma[x+1]/Gamma[x-n+1], {x, 0, 1}]]; Table[a[n], {n, 0, 19}] (* _Jean-François Alcover_, Jul 29 2013 *)
%t a[ n_] := If[ n < 0, 0, (-1)^n Numerator @ Integrate[ Pochhammer[ -x, n], {x, 0, 1}]]; (* _Michael Somos_, Jul 12 2014 *)
%t a[ n_] := If[ n < 0, 0, Numerator [ n! SeriesCoefficient[ x / Log[ 1 + x], {x, 0, n}]]]; (* _Michael Somos_, Jul 12 2014 *)
%t Join[{1}, Array[Numerator[(1/#) Integrate[Product[(x - k), {k, 0, # - 1}], {x, 0, 1}]] &, 25]] (* _Michael De Vlieger_, Nov 13 2018 *)
%o (Sage)
%o def A006232_list(len):
%o f, R, C = 1, [1], [1]+[0]*(len-1)
%o for n in (1..len-1):
%o for k in range(n, 0, -1):
%o C[k] = -C[k-1] * k / (k + 1)
%o C[0] = -sum(C[k] for k in (1..n))
%o R.append((C[0]*f).numerator())
%o f *= n
%o return R
%o print(A006232_list(20)) # _Peter Luschny_, Feb 19 2016
%o (PARI) for(n=0,20, print1(numerator( sum(k=0,n, stirling(n, k, 1)/(k+1)) ), ", ")) \\ _G. C. Greubel_, Nov 13 2018
%o (Magma) [Numerator((&+[StirlingFirst(n,k)/(k+1): k in [0..n]])): n in [0..20]]; // _G. C. Greubel_, Nov 13 2018
%o (Python) # Results are abs values
%o from fractions import gcd
%o aa,n,sden = [0,1],1,1
%o while n < 20:
%o j,snom,sden,a = 1,0,(n+1)*sden,0
%o while j < len(aa):
%o snom,j = snom+aa[j]*(sden//(j+1)),j+1
%o nom,den = snom,sden
%o print(n,nom//gcd(nom,den))
%o aa,j = aa+[-aa[j-1]],j-1
%o while j > 0:
%o aa[j],j = n*aa[j]-aa[j-1],j-1
%o n = n+1 # _A.H.M. Smeets_, Nov 14 2018
%o (Python)
%o from fractions import Fraction
%o from sympy.functions.combinatorial.numbers import stirling
%o def A006232(n): return sum(Fraction(stirling(n,k,kind=1,signed=True),k+1) for k in range(n+1)).numerator # _Chai Wah Wu_, Jul 09 2023
%Y Cf. A006233 (denominators), A002206, A002207, A002208, A002209, A002657, A002790.
%K sign,frac,nice,changed
%O 0,5
%A _N. J. A. Sloane_
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