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%I M4765 N2036 #78 Sep 08 2022 08:44:31
%S 1,1,11,5,137,7,363,761,7129,671,83711,6617,1145993,1171733,1195757,
%T 143327,42142223,751279,275295799,55835135,18858053,830139,444316699,
%U 269564591,34052522467,34395742267,312536252003,10876020307,9227046511387,300151059037
%N Numerator of the n-th harmonic number H(n) divided by (n+1); a(n) = A001008(n) / ((n+1)*A002805(n)).
%C Numerators of coefficients for numerical differentiation.
%D 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).
%D A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H M. F. Hasler, <a href="/A002547/b002547.txt">Table of n, a(n) for n = 1..2000</a> (first 700 terms from Alois P. Heinz)
%H W. G. Bickley and J. C. P. Miller, <a href="/A002551/a002551.pdf">Numerical differentiation near the limits of a difference table</a>, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]
%H A. N. Lowan, H. E. Salzer and A. Hillman, <a href="/A002545/a002545.pdf">A table of coefficients for numerical differentiation</a>, Bull. Amer. Math. Soc., 48 (1942), 920-924. [Annotated scanned copy]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HarmonicNumber.html">Harmonic Number</a>
%F G.f.: (-log(1-x))^2 (for fractions A002547(n)/A002548(n)). - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002
%F A002547(n)/A002548(n) = 2*Stirling_1(n+2, 2)(-1)^n/(n+2)! - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002
%F Numerator of u(n) = Sum_{k=1..n-1} 1/(k*(n-k)) (u(n) is asymptotic to 2*log(n)/n). - _Benoit Cloitre_, Apr 12 2003; corrected by _Istvan Mezo_, Oct 29 2012
%F a(n) = numerator of 2*Integral_{0..1} x^(n+1)*log(x/(1-x)) dx. - _Groux Roland_, May 18 2011
%F a(n) = numerator of A001008(n)/(n+1), since A001008(n)/A002805(n) are already in lowest terms. - _M. F. Hasler_, Jul 03 2019
%e H(n) = Sum_{k=1..n} 1/k, begins 1, 3/2, 11/6, 25/12, ... so H(n)/(n+1) begins 1/2, 1/2, 11/24, 5/12, ....
%e a(4) = numerator(H(4)/(4+1)) = 5.
%p H := proc(a, b) option remember; local m, p, q, r, s;
%p if b - a <= 1 then return 1, a fi; m := iquo(a + b, 2);
%p p, q := H(a, m); r, s := H(m, b); p*s + q*r, q*s; end:
%p A002547 := proc(n) H(1, n+1); numer(%[1]/(%[2]*(n+1))) end:
%p seq(A002547(n), n=1..30); # _Peter Luschny_, Jul 11 2019
%t a[n_]:= Numerator[HarmonicNumber[n]/(n+1)])]; Table[a[n], {n, 35}] (* modified by _G. C. Greubel_, Jul 03 2019 *)
%o (PARI) h(n) = sum(k=1, n, 1/k);
%o vector(35, n, numerator(h(n)/(n+1))) \\ _G. C. Greubel_, Jul 03 2019
%o (PARI) A002547(n)=numerator(A001008(n)/(n+1)) \\ _M. F. Hasler_, Jul 03 2019
%o (Magma) [Numerator(HarmonicNumber(n)/(n+1)): n in [1..35]]; // _G. C. Greubel_, Jul 03 2019
%o (Sage) [numerator(harmonic_number(n)/(n+1)) for n in (1..35)] # _G. C. Greubel_, Jul 03 2019
%o (GAP) List([1..35], n-> NumeratorRat(Sum([1..n], k-> 1/k)/(n+1))) # _G. C. Greubel_, Jul 03 2019
%Y Cf. A002548, A001008, A002805.
%K nonn,frac
%O 1,3
%A _N. J. A. Sloane_
%E More terms from Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002
%E Simpler definition from _Alexander Adamchuk_, Oct 31 2004
%E Offset corrected by _Gary Detlefs_, Sep 08 2011
%E Definition corrected by _M. F. Hasler_, Jul 03 2019
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