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A002548 Denominators of coefficients for numerical differentiation.

%I A002548 M4822 N2063 #57 Feb 16 2025 08:32:26
%S A002548 1,1,12,6,180,10,560,1260,12600,1260,166320,13860,2522520,2702700,
%T A002548 2882880,360360,110270160,2042040,775975200,162954792,56904848,
%U A002548 2586584,1427794368,892371480,116008292400,120470149800,1124388064800
%N A002548 Denominators of coefficients for numerical differentiation.
%C A002548 Denominator of 1 - 2*HarmonicNumber(n-1)/n. - _Eric W. Weisstein_, Apr 15 2004
%C A002548 Denominator 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
%C A002548 Expected area of the convex hull of n points picked at random inside a triangle with unit area. - _Eric W. Weisstein_, Apr 15 2004
%D A002548 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002548 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002548 Vincenzo Librandi, <a href="/A002548/b002548.txt">Table of n, a(n) for n = 2..250</a>
%H A002548 W. G. Bickley and J. C. P. Miller, <a href="http://dx.doi.org/10.1080/14786444208521334">Numerical differentiation near the limits of a difference table</a>, Phil. Mag., 33 (1942), 1-12 (plus tables).
%H A002548 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 A002548 A. N. Lowan, H. E. Salzer and A. Hillman, <a href="http://projecteuclid.org/euclid.bams/1183504875">A table of coefficients for numerical differentiation</a>, Bull. Amer. Math. Soc., 48 (1942), 920-924.
%H A002548 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 A002548 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TrianglePointPicking.html">Triangle Point Picking</a>
%H A002548 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SimplexSimplexPicking.html">Simplex Simplex Picking</a>
%H A002548 <a href="/index/J#Josephus">Index entries for sequences related to the Josephus Problem</a>
%F A002548 G.f.: (-log(1-x))^2 (for fractions A002547(n)/A002548(n)).
%F A002548 A002547(n)/a(n) = 2*Stirling_1(n+2, 2)(-1)^n/(n+2)!.
%e A002548 0, 0, 1/12, 1/6, 43/180, 3/10, 197/560, 499/1260, 5471/12600, ...
%p A002548 seq(denom(Stirling1(j+2,2)/(j+2)!*2!*(-1)^j), j=0..50);
%t A002548 Table[Denominator[1 - 2*HarmonicNumber[n - 1]/n], {n, 2, 30}] (* _Wesley Ivan Hurt_, Mar 24 2014 *)
%Y A002548 Cf. A002547, A093762.
%K A002548 nonn,frac
%O A002548 2,3
%A A002548 _N. J. A. Sloane_
%E A002548 More terms, GF, formula, Maple code from Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002
%E A002548 Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, Jun 16 2007