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 A006233 Denominators of Cauchy numbers of first type. (Formerly M1558) 23

%I M1558

%S 1,2,6,4,30,4,84,24,90,20,132,8,5460,840,360,48,1530,4,1596,168,1980,

%T 1320,8280,80,81900,6552,1512,112,3480,80,114576,7392,117810,7140,

%U 1260,8,3838380,5928,936,48,81180,440,1191960,55440,869400,38640,236880,224

%N Denominators of Cauchy numbers of first type.

%C 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 Cauchy numbers of the first type are also called Bernoulli numbers of the second kind.

%D A. Adelberg, 2-adic congruences of Norland numbers and of Bernoulli numbers of the second kind, J. Number Theory, 73 (1998), 47-58.

%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.

%D H. Jeffreys and B. S. Jeffreys, Methods of Mathematical Physics, Cambridge, 1946, p. 259.

%D Merlini, Donatella; Sprugnoli, Renzo; and Verri, M. Cecilia; The Cauchy numbers. Discrete Math. 306 (2006), no. 16, 1906-1920.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D Ming Wu and Hao Pan, Sums of products of Bernoulli numbers of the second kind, Fib. Quart., 45 (2007), 146-150.

%D Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., 309 (2009), 3830-3842.

%H T. D. Noe, <a href="/A006233/b006233.txt">Table of n, a(n) for n=0..1000</a>

%H W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A006232.text">Sheffer a- and z-sequences. </a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BernoulliNumberoftheSecondKind.html">Bernoulli Number of the Second Kind.</a>

%F Denominator of integral of x(x-1)...(x-n+1) from 0 to 1.

%F E.g.f.: x/log(1+x).

%F Denominator of Sum_{k=0..n} A048994(n,k)/(k+1). [From _Peter Luschny_, Apr 28 2009]

%e 1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90,...

%p seq(denom(add(stirling1(n,k)/(k+1),k=0..n)),n=0..12); [From _Peter Luschny_, Apr 28 2009]

%t With[{nn=50},Denominator[CoefficientList[Series[x/Log[1+x],{x,0,nn}],x] Range[0,nn]!]] (* From Harvey P. Dale, Oct 28 2011 *)

%t 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_ *)

%Y Cf. A006232, A002206, A002207, A002208, A002209, A002657, A002790.

%K nonn,frac,nice,easy

%O 0,2

%A _N. J. A. Sloane_.

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Last modified June 18 18:23 EDT 2013. Contains 226355 sequences.