A002548 Denominators of coefficients for numerical differentiation.

1, 1, 12, 6, 180, 10, 560, 1260, 12600, 1260, 166320, 13860, 2522520, 2702700, 2882880, 360360, 110270160, 2042040, 775975200, 162954792, 56904848, 2586584, 1427794368, 892371480, 116008292400, 120470149800, 1124388064800

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• frac

Comments

Denominator of 1 - 2*HarmonicNumber(n-1)/n. - Eric W. Weisstein, Apr 15 2004

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

Expected area of the convex hull of n points picked at random inside a triangle with unit area. - Eric W. Weisstein, Apr 15 2004

Examples

			0, 0, 1/12, 1/6, 43/180, 3/10, 197/560, 499/1260, 5471/12600, ...
		

References

• 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).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..250
• 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).
• 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) [Annotated scanned copy]
• A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924.
• A. N. Lowan, H. E. Salzer and A. Hillman, A table of coefficients for numerical differentiation, Bull. Amer. Math. Soc., 48 (1942), 920-924. [Annotated scanned copy]
• Eric Weisstein's World of Mathematics, Triangle Point Picking
• Eric Weisstein's World of Mathematics, Simplex Simplex Picking
• Index entries for sequences related to the Josephus Problem

Crossrefs

Cf. A002547, A093762.

Programs

Maple

seq(denom(Stirling1(j+2,2)/(j+2)!*2!*(-1)^j), j=0..50);

Mathematica

Table[Denominator[1 - 2*HarmonicNumber[n - 1]/n], {n, 2, 30}] (* Wesley Ivan Hurt, Mar 24 2014 *)

Formula

G.f.: (-log(1-x))^2 (for fractions A002547(n)/A002548(n)).

A002547(n)/a(n) = 2*Stirling_1(n+2, 2)(-1)^n/(n+2)!.

Extensions

More terms, GF, formula, Maple code from Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Jun 16 2007