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A002206 Numerators of logarithmic numbers (also of Gregory coefficients G(n)).
(Formerly M5066 N2194)
14
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/[(ln^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) [From Groux Roland, Jan 14 2009]

This formula for Euler's constant was discovered in 1924 by the Dutch mathematician Jan C. Kluyver (1860-1932). - Hans J. H. Tuenter, Feb 26 2012

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..inf} |G(n)|/(n-1) = 1/2*(log(2*Pi) - 1 - euler_gamma) and

sum {n = 1..inf} |G(n)|/(n+1) = 1 - log(2).

(End)

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 formulae, 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).

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

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

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.

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

Eric Weisstein's World of Mathematics, Logarithmic Number

Wikipedia, Euler-Maclaurin formula

Wikipedia, Euler-Mascheroni constant: Relations with the reciprocal logarithm.

Index entries for sequences related to logarithmic numbers

FORMULA

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

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! * 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

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(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-Fran├žois Alcover, Nov 29 2013, after Vladimir Kruchinin *)

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]

CROSSREFS

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

Sequence in context: A040353 A128160 A092120 * A040349 A040350 A089572

Adjacent sequences:  A002203 A002204 A002205 * A002207 A002208 A002209

KEYWORD

sign,frac,nice

AUTHOR

N. J. A. Sloane.

STATUS

approved

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