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Search: id:A002547
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| A002547 |
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Numerator of {n-th harmonic number H(n) divided by (n+1)}: a(n) = Numerator[HarmonicNumber[n]/(n+1)], H(n) = HarmonicNumber[n] = A001008(n)/A002805(n).
(Formerly M4765 N2036)
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+0
3
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| 1, 1, 11, 5, 137, 7, 363, 761, 7129, 671, 83711, 6617, 1145993, 1171733, 1195757, 143327, 42142223, 751279, 275295799, 55835135, 18858053, 830139, 444316699, 269564591, 34052522467, 34395742267, 312536252003, 10876020307, 9227046511387, 300151059037
(list; graph; listen)
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OFFSET
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2,3
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COMMENT
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Numerators of coefficients for numerical differentiation.
Numerator of u(n)=sum(k=1,n,1/k/(n-k)) (u(n) is asymptotic to 2*log(n)/n) - Benoit Cloitre (benoit7848c(AT)orange.fr), Apr 12 2003
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REFERENCES
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W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924.
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LINKS
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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
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FORMULA
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G.f.: (-ln(1-x))^2 (for fractions A002547(n)/A002548(n))
A002547(n)/A002548(n)=2 stirling1(n+2, n)(-1)^n/(n+2)!
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EXAMPLE
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E.g. H(n) = Sum[1/i,{i,1,n}] begins 1, 3/2, 11/6, 25/12, ... so H(n)/(n+1) begins 1/2, 1/2, 11/24, 5/12, ... so a(4) = 5.
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MAPLE
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with(combinat):seq(numer(stirling1(j+2, 2)/(j+2)!*2!*(-1)^j), j=0..50);
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MATHEMATICA
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Numerator[HarmonicNumber[n]/(n+1)]
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CROSSREFS
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Cf. A002548.
Cf. A001008, A002805.
Sequence in context: A127820 A120831 A038319 this_sequence A090840 A080501 A122098
Adjacent sequences: A002544 A002545 A002546 this_sequence A002548 A002549 A002550
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KEYWORD
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nonn,frac
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AUTHOR
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njas
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EXTENSIONS
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More terms, GF, formula, Maple code from Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19, 2002
Simpler definition from Alexander Adamchuk (alex(AT)kolmogorov.com), Oct 31 2004
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