OFFSET
-1,5
COMMENTS
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
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 = 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
REFERENCES
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].
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).
LINKS
T. D. Noe, Table of n, a(n) for n=-1..100
Ibrahim M. Alabdulmohsin, "The Language of Finite Differences", in Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Springer, Cham, 2018, pp. 133-149.
Ibrahim M. Alabdulmohsin, Summability Calculus, arXiv:1209.5739 [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], 2014.
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.
Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.
Iaroslav V. Blagouchine and Marc-Antoine Coppo, A note on some constants related to the zeta-function and their relationship with the Gregory coefficients, arXiv:1703.08601 [math.NT], 2017. Also The Ramanujan Journal 47.2 (2018): 457-473.
Mark W. Coffey and Jonathan Sondow, Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, Acta Appl. Math., Vol. 121 (2012), pp. 1-3.
J. C. Kluyver, Euler's constant and natural numbers, Proc. K. Ned. Akad. Wet., Vol. 27, No. 1-2 (1924), pp. 142-144.
Victor Kowalenko, Properties and Applications of the Reciprocal Logarithm Numbers, Acta Applic. Mathem. 109 (2) (2010) 413-437.
Arnold N. Lowan and Herbert E. Salzer, Table of coefficients in numerical integration formulas, J. Math. Phys. Mass. Inst. Tech., Vol. 22 (1943), pp. 49-50.[Annotated scanned copy]
Toshiki Matsusaka, Hideki Murahara, and Tomokazu Onozuka, Asymptotic coefficients of multiple zeta functions at the origin and generalized Gregory coefficients, arXiv:2312.14475 [math.NT], 2023.
Gergő Nemes, An Asymptotic Expansion for the Bernoulli Numbers of the Second Kind, J. Int. Seq. 14 (2011) # 11.4.8.
G. M. Phillips, Gregory's method for numerical integration, Amer. Math. Monthly, Vol. 79, No. 3 (1972), pp. 270-274.
Herbert E. Salzer, Table of coefficients for repeated integration with differences, Phil. Mag., Vol. 38 (1947), pp. 331-336. [Annotated scanned copy]
Raphael Schumacher, Rapidly Convergent Summation Formulas involving Stirling Series, arXiv preprint arXiv:1602.00336, 2016
Patricia C. Stamper, Table of Gregory coefficients, Math. Comp., Vol. 20, No. 95 (1966), p. 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., Vol. 45, No. 2 (2007), pp. 146-150.
FORMULA
1/log(1+x) = Sum_{n>=-1} (a(n)/A002207(n)) * x^n. [corrected by Robert Israel, Oct 22 2015]
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
a(n)/A002207(n) = 1/(n+1)! * Sum_{k=0..n+1} Stirling1(n+1,k)/(k+1). - Vladimir Kruchinin, Sep 23 2012
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)/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
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
EXAMPLE
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 *)
Numerator@CoefficientList[x/Log[1+x] + O[x]^21, x] (* Oliver Seipel, Jul 06 2024 *)
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 */
(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
CROSSREFS
KEYWORD
sign,frac,nice
AUTHOR
STATUS
approved