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A002657
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Numerators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).
(Formerly M3790 N1545)
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14
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1, 1, 5, 9, 251, 475, 19087, 36799, 1070017, 2082753, 134211265, 262747265, 703604254357, 1382741929621, 8164168737599, 5362709743125, 8092989203533249, 15980174332775873, 12600467236042756559
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| These coefficients (with alternating signs) are also known as the Nørlund [or Norlund or Noerlund] numbers.
The denominators are found in A002790. The alternating rational sequence ((-1)^n)*a(n)/A002790(n)is the z-sequence for the Stirling2 triangle A008277(n+1,k+1),n>=k>=0. This is the Sheffer (exp(x),exp(x)-1) triangle. See the W. Lang link under A006232 for Sheffer a- and z-sequences with references, and the conversion to S. Roman's notation. The a-sequence is A006232(n)/A006233(n). [Wolfdieter Lang, Oct 06 2011]
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REFERENCES
| L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
P. Curtz, Integration numerique des systemes differentiels .., C.C.S.A.,Arcueil,1969. [From Paul Curtz (bpcrtz(AT)free.fr), Nov 24 2008]
Guodong Liu, Some computational formulas for Norlund numbers, Fib. Quart., 45 (2007), 133-137.
Merlini, Donatella; Sprugnoli, Renzo; and Verri, M. Cecilia; The Cauchy numbers. Discrete Math. 306 (2006), no. 16, 1906-1920.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 136.
N. E. Nørlund, Vorlesungen ueber Differenzenrechnung, Springer-Verlag, Berlin, 1924.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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
Index entries for sequences related to Bernoulli numbers.
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FORMULA
| Numerator of integral of x(x+1)...(x+n-1) from 0 to 1.
E.g.f.: -x/(1-x)/ln(1-x).
Numerator of Sum_{k=0..n} (-1)^(n-k) A008275(n,k)/(k+1) [From Peter Luschny (peter(AT)luschny.de), Apr 28 2009]
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EXAMPLE
| 1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ...
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MAPLE
| seq(numer(add((-1)^(n-k)*stirling1(n, k)/(k+1), k=0..n)), n=0..10); [From Peter Luschny (peter(AT)luschny.de), Apr 28 2009]
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MATHEMATICA
| Table[Abs[Numerator[NorlundB[n, n]]], {n, 0, 30}](*From Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), 30 Dec 2010*)
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CROSSREFS
| Cf. A002790. See also A002208, A002209, A002206, A002207, A006232, A006233.
Sequence in context: A097397 A092584 A145400 * A046093 A097086 A109076
Adjacent sequences: A002654 A002655 A002656 * A002658 A002659 A002660
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KEYWORD
| nonn,frac,easy,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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