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A002206
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Numerators of logarithmic numbers (also of Gregory coefficients G(n)).
(Formerly M5066 N2194)
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34
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1, 1, -1, 1, -19, 3, -863, 275, -33953, 8183, -3250433, 4671, -13695779093, 2224234463, -132282840127, 2639651053, -111956703448001, 50188465, -2334028946344463, 301124035185049, -12365722323469980029
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OFFSET
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-1,5
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COMMENTS
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For n>0 G(n) = (-1)^(n+1) * Integral_{x=0..infinity} 1/((log^2(x)+Pi^2)*(x+1)^n). G(1)=1/2, and for n>1, G(n) = (-1)^(n+1)/(n+1) - Sum_{k=1..n-1} (-1)^k*G(n-k)/(k+1). Euler's constant is given by gamma = Sum_{n>=1} (-1)^(n+1)*G(n)/n. - 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
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 = Integral_{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} 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
Named after the Scottish mathematician and astronomer James Gregory (1638-1675). - Amiram Eldar, Jun 16 2021
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REFERENCES
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Eugene Isaacson and Herbert Bishop Keller, 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.
Murray S. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990, see page 101 [Problem 87-6].\
Victor Kowalenko, Properties and Applications of the Reciprocal Logarithm Numbers, Acta Applic. Mathem. 109 (2) (2010) 413-437 doi:10.1007/s10440-008-9325-0
Arnold N. Lowan and Herbert E. Salzer, Table of coefficients in numerical integration formulas, J. Math. Phys. Mass. Inst. Tech. 22 (1943), 49-50.
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).
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LINKS
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FORMULA
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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
G(n) = (Integral_{x=0..1} x*(x-n)_n)/(n+1)!, where (a)_n is the Pochhammer symbol. - Vladimir Reshetnikov, Oct 22 2015
a(n) = numerator(f(n+1)), where f(0) = 1, f(n) = Sum_{k=0..n-1} (-1)^(n-k+1) * f(k) / (n-k+1). - Daniel Suteu, Nov 15 2018
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EXAMPLE
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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
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MAPLE
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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);
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MATHEMATICA
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Numerator@Table[Integrate[x Pochhammer[x - n, n], {x, 0, 1}]/(n + 1)!, {n, -1, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)
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PROG
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(Maxima) a(n):=sum(stirling1(n+1, k)/((n+1)!*(k+1)), k, 0, n+1);
(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 */
(PARI) a(n) = numerator(sum(k=0, n+1, stirling(n+1, k, 1)/((n+1)!*(k+1)))); \\ Michel Marcus, Mar 20 2018
(Python)
from math import factorial
from fractions import Fraction
from sympy.functions.combinatorial.numbers import stirling
def A002206(n): return (sum(Fraction(stirling(n+1, k, kind=1, signed=True), k+1) for k in range(n+2))/factorial(n+1)).numerator # Chai Wah Wu, Feb 12 2023
(SageMath)
from functools import cache
@cache
def h(n):
return (-sum((-1)**k * h(n - k) / (k + 1) for k in range(1, n + 1))
+ (-1)**n * n / (2*(n + 1)*(n + 2)))
def a(n): return h(n).numer() if n > 0 else 1
print([a(n) for n in range(-1, 20)]). # Peter Luschny, Dec 12 2023
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CROSSREFS
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KEYWORD
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sign,frac,nice
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AUTHOR
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STATUS
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approved
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