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A006232
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Numerators of Cauchy numbers of first type.
(Formerly M5067)
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44
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1, 1, -1, 1, -19, 9, -863, 1375, -33953, 57281, -3250433, 1891755, -13695779093, 24466579093, -132282840127, 240208245823, -111956703448001, 4573423873125, -30342376302478019, 56310194579604163
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| -a(n+1), n>=0, also numerators from e.g.f. 1/x-1/ln(1+x), with denominators A075178(n). |a(n+1)|, n>=0, numerators from e.g.f. 1/x+1/ln(1-x) with denominators A075178(n). For formula of unsigned a(n) see A075178.
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.
Cauchy numbers of the first type are also called Bernoulli numbers of the second kind.
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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.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..100
W. Lang, Sheffer a- and z-sequences.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
| Numerator of integral of x(x-1)...(x-n+1) from 0 to 1.
E.g.f.: x/log(1+x).
Numerator of Sum_{k=0..n} A048994(n,k)/(k+1). [From Peter Luschny (peter(AT)luschny.de), Apr 28 2009]
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EXAMPLE
| 1, 1/2, -1/6, 1/4, -19/30, 9/4, -863/84, 1375/24, -33953/90,...
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MAPLE
| seq(numer(add(stirling1(n, k)/(k+1), k=0..n)), n=0..20); [From Peter Luschny (peter(AT)luschny.de), Apr 28 2009]
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MATHEMATICA
| a[n_] := Numerator[ Sum[ StirlingS1[n, k]/(k + 1), {k, 0, n}]]; Table[a[n], {n, 0, 19}] (* From Jean-François Alcover, Nov 03 2011, after Maple *)
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CROSSREFS
| Cf. A006233, A002206, A002207, A002208, A002209, A002657, A002790.
Sequence in context: A033339 A175674 A175675 * A122549 A039942 A050276
Adjacent sequences: A006229 A006230 A006231 * A006233 A006234 A006235
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KEYWORD
| sign,frac,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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