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A002207 Denominators of logarithmic numbers (also of Gregory coefficients G(n)).
(Formerly M2017 N0797)
17
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. [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.

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

T. D. Noe, Table of n, a(n) for n=-1..100

M. Alabdulmohsin, Summability Calculus, arXiv:1209.5739v1 [math.CA], 2012.

Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. arXiv version, arXiv:1401.3724 [math.NT], 2004.

Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.

Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.

M. Coffey and J. Sondow, Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, Acta Appl. Math., 121 (2012), 1-3.

J. C. Kluyver, Euler's constant and natural numbers, Proc. K. Ned. Akad. Wet., 27(1-2) (1924), 142-144.

A. N. Lowan and H. Salzer, Table of coefficients in numerical integration formulas, J. Math. Phys., 22 (1943), 49-50.

A. N. Lowan and H. Salzer, Table of coefficients in numerical integration formulas, J. Math. Phys. Mass. Inst. Tech. 22 (1943), 49-50.[Annotated scanned copy]

G. M. Phillips, Gregory's method for numerical integration, Amer. Math. Monthly, 79 (1972), 270-274.

H. E. Salzer, Table of coefficients for repeated integration with differences, Phil. Mag., 38 (1947), 331-336.

H. E. Salzer, Table of coefficients for repeated integration with differences,  Phil. Mag., 38 (1947), 331-336. [Annotated scanned copy]

P. C. Stamper, Table of Gregory coefficients, Math. Comp., 20 (1966), 465.

Eric Weisstein's World of Mathematics, Logarithmic Number.

Wikipedia, Gregory coefficients

Ming Wu and Hao Pan, Sums of products of Bernoulli numbers of the second kind, Fib. Quart., 45 (2007), 146-150.

Index entries for sequences related to logarithmic numbers

FORMULA

1/log(1+x) = Sum_{n>=-1} (A002206(n)/a(n)) * x^n.

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

G(n) = integrate(x=0..1, x*(x-n)_n)/(n+1)!, where (a)_n is the Pochhammer symbol. - Vladimir Reshetnikov, Oct 22 2015

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

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 *)

Denominator@Table[Integrate[x Pochhammer[x - n, n], {x, 0, 1}]/(n + 1)!, {n, -1, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)

CROSSREFS

Cf. A002206, A006232, A006233, A002208, A002209, A002657, A002790, A195189, A270857, A270859.

Sequence in context: A141900 A211374 A126962 * A181814 A232248 A091137

Adjacent sequences:  A002204 A002205 A002206 * A002208 A002209 A002210

KEYWORD

nonn,frac,nice,changed

AUTHOR

N. J. A. Sloane.

STATUS

approved

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Last modified June 26 22:04 EDT 2016. Contains 274242 sequences.