%I M2651 N1058 #42 Nov 17 2018 20:45:11

%S 1,3,7,15,29,469,29531,1303,16103,190553,128977,9061,30946717,

%T 39646461,58433327,344499373,784809203,169704792667,665690574539,

%U 5667696059,337284946763,7964656853269,46951444927823,284451446729,1597747168263479,816088653136373

%N Numerator of Sum_{i+j+k=n; i,j,k > 0} 1/(i*j*k).

%C Numerators of coefficients for numerical differentiation.

%C a(n)/A002546(n) = 3*int(x^(n-1)*log^2(x/(1-x)),x=0..1)-(Pi^2)/n. - _Groux Roland_, Nov 13 2009

%C For prime p >= 5, a(p) == -2*Bernoulli(p-3) (mod p). (See Zhao link.) - _Michel Marcus_, Feb 05 2016

%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 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 Jianqiang Zhao, <a href="http://dx.doi.org/10.1016/j.jnt.2006.05.005">Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem</a>, Journal of Number Theory, Volume 123, Issue 1, March 2007, Pages 18-26. See Corollary 4.2 p. 25.

%F G.f.: (-log(1-x))^3 (for fractions A002545(n)/A002546(n)). - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

%F A002545(n)/A002546(n) = 6*Stirling_1(n+3, 3)*(-1)^n/(n+3)!. - Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

%p seq(numer(-Stirling1(j, 3)/j!*3!*(-1)^j), j=3..50); # Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002

%t Table[Sum[1/i/j/(n-i-j), {i, n-2}, {j, n-i-1}], {n, 3, 100}] (* _Ryan Propper_ *)

%Y Cf. A002546.

%K nonn,frac

%O 3,2

%A _N. J. A. Sloane_

%E More terms from Barbara Margolius (b.margolius(AT)math.csuohio.edu), Jan 19 2002