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A006233 Denominators of Cauchy numbers of first type.
(Formerly M1558)
27
1, 2, 6, 4, 30, 4, 84, 24, 90, 20, 132, 8, 5460, 840, 360, 48, 1530, 4, 1596, 168, 1980, 1320, 8280, 80, 81900, 6552, 1512, 112, 3480, 80, 114576, 7392, 117810, 7140, 1260, 8, 3838380, 5928, 936, 48, 81180, 440, 1191960, 55440, 869400, 38640, 236880, 224 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

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.

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

REFERENCES

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

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

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

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

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

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

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

LINKS

T. D. Noe, Table of n, a(n) for n=0..1000

W. Lang, Sheffer a- and z-sequences.

Eric Weisstein's World of Mathematics, Bernoulli Number of the Second Kind.

FORMULA

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

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

Denominator of Sum_{k=0..n} A048994(n,k)/(k+1). [From Peter Luschny, Apr 28 2009]

EXAMPLE

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

MAPLE

seq(denom(add(stirling1(n, k)/(k+1), k=0..n)), n=0..12); [From Peter Luschny, Apr 28 2009]

MATHEMATICA

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

a[n_] := Sum[ StirlingS1[n, k]/(k+1), {k, 0, n}] // Denominator; Table[a[n], {n, 0, 40}] (* Jean-Fran├žois Alcover, Jan 10 2013, after Peter Luschny *)

CROSSREFS

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

Sequence in context: A038212 A039656 A226532 * A164020 A057643 A073039

Adjacent sequences:  A006230 A006231 A006232 * A006234 A006235 A006236

KEYWORD

nonn,frac,nice,easy

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified October 30 11:14 EDT 2014. Contains 248797 sequences.