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A002206 Numerators of logarithmic numbers (also of Gregory coefficients G(n)).
(Formerly M5066 N2194)
26
1, 1, -1, 1, -19, 3, -863, 275, -33953, 8183, -3250433, 4671, -13695779093, 2224234463, -132282840127, 2639651053, -111956703448001, 50188465, -2334028946344463, 301124035185049, -12365722323469980029 (list; graph; refs; listen; history; text; internal format)
OFFSET

-1,5

COMMENTS

For n>0 G(n)=(-1)^(n+1)*int(1/[(log^2(x)+Pi^2)*(x+1)^n],x=0..infinity). G(1)=1/2[ for n>1 G(n)=(-1)^(n+1)/(n+1)-sum((-1)^k*G(n-k)/(k+1),k=1..n-1). Euler's constant is given by gamma=sum((-1)^(n+1)*G(n)/n,n=1..infinity). - Groux Roland, Jan 14 2009

The above series for Euler's constant was discovered circa 1780-1790 by the Italian mathematicians Gregorio Fontana (1735-1803) and Lorenzo Mascheroni (1750-1800), and was subsequently rediscovered several times (in particular, by Ernst Schröder in 1879, Niels E. Nørlund in 1923, Jan C. Kluyver in 1924, Charles Jordan in 1929, Kenter in 1999, and Victor Kowalenko in 2008). For more details, see references [Blagouchine, 2015] and [Blagouchine, 2016] below. - Iaroslav V. Blagouchine, Sep 16 2015

From Peter Bala, Sep 28 2012: (Start)

Gregory's coefficients {G(n)}n>=0 = {1,1/2,-1/12,1/24,-19/720,3/160,...} occur in Gregory's quadrature formula for numerical integration. The integral I = int {x = m..n} f(x) dx may be approximated by the sum S = 1/2*f(m) + f(m+1) + ... + f(n-1) + 1/2*f(n). Gregory's formula for the difference is

I - S = sum {k = 2..inf} G(k)*{delta^(k-1)(f(n)) - delta^(k-1)(f(m))}, where delta is the difference operator delta(f(x)) = f(x+1) - f(x).

Gregory's formula is the discrete analog of the Euler-Maclaurin summation formula, with finite differences replacing derivatives and the Gregory coefficients replacing the Bernoulli numbers.

Alabdulmohsin, Section 7.3.3, gives several identities involving the Gregory coefficients including

Sum_{n >= 2} |G(n)|/(n-1) = (1/2)*(log(2*Pi) - 1 - euler_gamma) and

Sum_{n >= 1} |G(n)|/(n+1) = 1 - log(2).

(End)

More series with Gregory coefficients, accurate bounds for them, their full asymptotics at large index, as well as many historical details related to them, are given in the articles by Blagouchine (see refs. below). - Iaroslav V. Blagouchine,  May 06 2016

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.

M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990, see page 101 [Problem 87-6].

A. N. Lowan and H. Salzer, Table of coefficients in numerical integration formulas, J. Math. Phys. Mass. Inst. Tech. 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).

LINKS

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

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

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. 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. [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} (a(n)/A002207(n)) * x^n.

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)).

a(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), Jan 21 2002

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

a(n)/A002207(n) = (1/n!)*Sum_{k=0..n+1} (-1)^(k+1)*stirling2(n+k+1,k)* binomial(2*n+1,n+k)/((n+k+1)*(n+k)), n>0, with a(-1)/A002207(-1)=1,  a(0)/A002207(0)=1/2. - Vladimir Kruchinin, Apr 05 2016

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(numer(1/i!*sum(bernoulli(j)/(j)*stirling1(i, j-1), j=1..i+1)), i=1..24);

MATHEMATICA

a[n_] := Sum[StirlingS1[n+1, k]/((n+1)!*(k+1)), {k, 0, n+1}]; Table[a[n] // Numerator, {n, -1, 19}] (* Jean-François Alcover, Nov 29 2013, after Vladimir Kruchinin *)

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

PROG

(Maxima) a(n):=sum(stirling1(n+1, k)/((n+1)!*(k+1)), k, 0, n+1);

makelist(num(a(n)), n, -1, 10); /* Vladimir Kruchinin, Sep 23 2012 */

(Maxima)

a(n):=if n=-1 then 1 else if n=0 then 1/2 else  1/n!*sum(((-1)^(k+1)*stirling2(n+k+1, k)*binomial(2*n+1, n+k))/((n+k+1)*(n+k)), k, 0, n+1); /* Vladimir Kruchinin, Apr 05 2016 */

CROSSREFS

Cf. A001620, A002207, A006232, A006233, A002208, A002209, A002657, A002790.

Sequence in context: A040353 A128160 A092120 * A040349 A274249 A040350

Adjacent sequences:  A002203 A002204 A002205 * A002207 A002208 A002209

KEYWORD

sign,frac,nice,changed

AUTHOR

N. J. A. Sloane

EXTENSIONS

First formula corrected by Robert Israel, Oct 22 2015

STATUS

approved

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Last modified June 28 18:53 EDT 2016. Contains 274271 sequences.