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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; text; 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, Logarithmic Number.

Index entries for sequences related to logarithmic numbers

FORMULA

G.f.: 1/log(1+x).

A002206(n)/A002207(n) = 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)).

A002206(n)/A002207(n) = 1/(n+1)!*sum(k=0..n+1, stirling1(n+1,k)/(k+1)). - Vladimir Kruchinin, Sep 23 2012

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}] (* John M. Campbell, Dec 01 2011 *)

a[n_] := Denominator[n!^-1*Sum[BernoulliB[j]/j*StirlingS1[n, j-1], {j, 1, n+1}]]; a[-1] = 1; Table[a[n], {n, -1, 18}] (* Jean-Fran├žois Alcover, May 16 2012, after Maple *)

CROSSREFS

Cf. A002206, A006232, A006233, A002208, A002209, A002657, A002790.

Sequence in context: A141900 A211374 A126962 * A181814 A232248 A091137

Adjacent sequences:  A002204 A002205 A002206 * A002208 A002209 A002210

KEYWORD

nonn,frac,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified September 2 02:37 EDT 2014. Contains 246321 sequences.