|
| |
|
|
A002207
|
|
Denominators of logarithmic numbers (also of Gregory coefficients G(n)).
(Formerly M2017 N0797)
|
|
14
| |
|
|
1, 2, 12, 24, 720, 160, 60480, 24192, 3628800, 1036800, 479001600, 788480, 2615348736000, 475517952000, 31384184832000, 689762304000, 32011868528640000, 15613165568, 786014494949376000, 109285437800448000
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| -1,2
|
|
|
COMMENTS
| Denominator of the determinant of the (n+1)X(n+1) matrix with 1's along the superdiagonal, (1/2)'s along the main diagonal, (1/3)'s along the subdiagonal, etc., and 0's everywhere else. [From John M. Campbell, Dec 01 2011]
|
|
|
REFERENCES
| E. Isaacson and H. Bishop, Analysis of Numerical Methods, ISBN 0 471 42865 5, 1966, John Wiley and Sons, pp. 318-319 - Rudi Huysmans (rudi_huysmans(AT)hotmail.com), Apr 10 2000
Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 266.
A. N. Lowan and H. Salzer, Table of coefficients in numerical integration formulae, J. Math. Phys., 22 (1943), 49-50.
H. E. Salzer, Table of coefficients for repeated integration with differences, Phil. Mag., 38 (1947), 331-336.
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).
P. C. Stamper, Table of Gregory coefficients, Math. Comp., 20 (1966), 465.
Ming Wu and Hao Pan, Sums of products of Bernoulli numbers of the second kind, Fib. Quart., 45 (2007), 146-150.
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=-1..100
G. M. Phillips, Gregory's method for numerical integration, Amer. Math. Monthly, 79 (1972), 270-274.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for sequences related to logarithmic numbers
|
|
|
FORMULA
| G.f.: 1/log(1+x).
a(n)=A002206(n)/A002207=1/n! sum_{j=1}^{n+1} bernoulli(j)/j S_1(n, j-1), where S_1(n, k) is the Stirling number of the first kind. - Barbara Margolius (b.margolius(AT)csuohio.edu), 1/21/02
G(0)=0, G(n)=Sum_{i=1..n} (-1)^(i+1)*G(n-i)/(i+1)+(-1)^(n+1)*n/(2*(n+1)*(n+2)).
|
|
|
EXAMPLE
| Logarithmic numbers are 1, 1/2, -1/12, 1/24, -19/720, 3/160, -863/60480, 275/24192, -33953/3628800, 8183/1036800, -3250433/479001600, 4671/788480, -13695779093/2615348736000, 2224234463/475517952000, ... = A002206/A002207
G(0), G(1), ... = 0, 1/2, -1/12, 1/24, -19/720, 3/160, -863/60480, 275/24192, -33953/3628800, 8183/1036800, -3250433/479001600, 4671/788480, -13695779093/2615348736000, 2224234463/475517952000, ... = A002206/A002207
|
|
|
MAPLE
| series(1/log(1+x), x, 25);
with(combinat, stirling1):seq(denom(1/i!*sum(bernoulli(j)/(j)*stirling1(i, j-1), j=1..i+1)), i=1..24);
|
|
|
MATHEMATICA
| Table[Denominator[Det[Array[Sum[KroneckerDelta[#1, #2+q]*1/(q+2)^1, {q, -1, n+1}] &, {n+1, n+1}]]], {n, 0, 20}] (* From John M. Campbell, Dec 01 2011 *)
|
|
|
CROSSREFS
| Cf. A002206, A006232, A006233, A002208, A002209, A002657, A002790.
Sequence in context: A176710 A141900 A126962 * A181814 A091137 A092825
Adjacent sequences: A002204 A002205 A002206 * A002208 A002209 A002210
|
|
|
KEYWORD
| nonn,frac,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
