A002545 Numerator of Sum_{i+j+k=n; i,j,k > 0} 1/(i*j*k). (Formerly M2651 N1058)

1, 3, 7, 15, 29, 469, 29531, 1303, 16103, 190553, 128977, 9061, 30946717, 39646461, 58433327, 344499373, 784809203, 169704792667, 665690574539, 5667696059, 337284946763, 7964656853269, 46951444927823, 284451446729, 1597747168263479, 816088653136373

Offset

3,2

Comments

Numerators of coefficients for numerical differentiation.

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

For prime p >= 5, a(p) == -2*Bernoulli(p-3) (mod p). (See Zhao link.) - Michel Marcus, Feb 05 2016

References

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.

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

Table of n, a(n) for n=3..28.

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. [Annotated scanned copy]

Jianqiang Zhao, Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem, Journal of Number Theory, Volume 123, Issue 1, March 2007, Pages 18-26. See Corollary 4.2 p. 25.

Formula

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

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

Maple

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

Mathematica

Table[Sum[1/i/j/(n-i-j), {i, n-2}, {j, n-i-1}], {n, 3, 100}] (* Ryan Propper *)

Crossrefs

Cf. A002546.

Sequence in context: A283864 A283606 A147400 * A153114 A325664 A290865

Adjacent sequences: A002542 A002543 A002544 * A002546 A002547 A002548

Keyword

nonn,frac

Author

N. J. A. Sloane

Extensions

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

Status

approved