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%I M5066 N2194 #174 Dec 29 2023 11:13:06
%S 1,1,-1,1,-19,3,-863,275,-33953,8183,-3250433,4671,-13695779093,
%T 2224234463,-132282840127,2639651053,-111956703448001,50188465,
%U -2334028946344463,301124035185049,-12365722323469980029
%N Numerators of logarithmic numbers (also of Gregory coefficients G(n)).
%C 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
%C 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
%C From _Peter Bala_, Sep 28 2012: (Start)
%C 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).
%C 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.
%C Alabdulmohsin, Section 7.3.3, gives several identities involving the Gregory coefficients including
%C Sum_{n >= 2} |G(n)|/(n-1) = (1/2)*(log(2*Pi) - 1 - euler_gamma) and
%C Sum_{n >= 1} |G(n)|/(n+1) = 1 - log(2).
%C (End)
%C 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
%C Named after the Scottish mathematician and astronomer James Gregory (1638-1675). - _Amiram Eldar_, Jun 16 2021
%D 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
%D Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 266.
%D Murray S. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990, see page 101 [Problem 87-6].\
%D 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
%D Arnold N. Lowan and Herbert E. Salzer, Table of coefficients in numerical integration formulas, J. Math. Phys. Mass. Inst. Tech. 22 (1943), 49-50.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002206/b002206.txt">Table of n, a(n) for n=-1..100</a>
%H Ibrahim M. Alabdulmohsin, <a href="https://doi.org/10.1007/978-3-319-74648-7_7">"The Language of Finite Differences"</a>, in Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Springer, Cham, 2018, pp. 133-149.
%H Ibrahim M. Alabdulmohsin, <a href="https://arxiv.org/abs/1209.5739">Summability Calculus</a>, arXiv:1209.5739 [math.CA], 2012.
%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jnt.2014.08.009">A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations</a>, Journal of Number Theory (Elsevier), vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. <a href="http://arxiv.org/abs/1401.3724">arXiv version</a>, arXiv:1401.3724 [math.NT], 2014.
%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jnt.2015.06.012">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</a>, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. <a href="http://arxiv.org/abs/1501.00740">arXiv version</a>, arXiv:1501.00740 [math.NT], 2015.
%H Iaroslav V. Blagouchine, <a href="http://dx.doi.org/10.1016/j.jmaa.2016.04.032">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</a>, Journal of Mathematical Analysis and Applications (Elsevier), 2016. <a href="http://arxiv.org/abs/1408.3902">arXiv version</a>, arXiv:1408.3902 [math.NT], 2014-2016.
%H Iaroslav V. Blagouchine, <a href="http://math.colgate.edu/~integers/sjs3/sjs3.Abstract.html">Three notes on Ser's and Hasse's representation for the zeta-functions</a>, Integers (2018) 18A, Article #A3.
%H Iaroslav V. Blagouchine and Marc-Antoine Coppo, <a href="https://arxiv.org/abs/1703.08601">A note on some constants related to the zeta-function and their relationship with the Gregory coefficients</a>, arXiv:1703.08601 [math.NT], 2017. Also The Ramanujan Journal 47.2 (2018): 457-473.
%H Mark W. Coffey and Jonathan Sondow, <a href="http://arxiv.org/abs/1202.3093">Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant</a>, Acta Appl. Math., Vol. 121 (2012), pp. 1-3.
%H J. C. Kluyver, <a href="http://www.dwc.knaw.nl/DL/publications/PU00015025.pdf">Euler's constant and natural numbers</a>, Proc. K. Ned. Akad. Wet., Vol. 27, No. 1-2 (1924), pp. 142-144.
%H Arnold N. Lowan and Herbert E. Salzer, <a href="/A002206/a002206.pdf">Table of coefficients in numerical integration formulas</a>, J. Math. Phys. Mass. Inst. Tech., Vol. 22 (1943), pp. 49-50.[Annotated scanned copy]
%H Toshiki Matsusaka, Hideki Murahara, and Tomokazu Onozuka, <a href="https://arxiv.org/abs/2312.14475">Asymptotic coefficients of multiple zeta functions at the origin and generalized Gregory coefficients</a>, arXiv:2312.14475 [math.NT], 2023.
%H Gergő Nemes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Nemes/nemes4.html">An Asymptotic Expansion for the Bernoulli Numbers of the Second Kind</a>, J. Int. Seq. 14 (2011) # 11.4.8.
%H G. M. Phillips, <a href="http://links.jstor.org/sici?sici=0002-9890%28197203%2979%3A3%3C270%3AGMFNI%3E2.0.CO%3B2-S">Gregory's method for numerical integration</a>, Amer. Math. Monthly, Vol. 79, No. 3 (1972), pp. 270-274.
%H Herbert E. Salzer, <a href="/A002206/a002206_1.pdf">Table of coefficients for repeated integration with differences</a>, Phil. Mag., Vol. 38 (1947), pp. 331-336. [Annotated scanned copy]
%H Raphael Schumacher, <a href="http://arxiv.org/abs/1602.00336">Rapidly Convergent Summation Formulas involving Stirling Series</a>, arXiv preprint arXiv:1602.00336, 2016
%H Patricia C. Stamper, <a href="http://dx.doi.org/10.1090/S0025-5718-66-99917-0">Table of Gregory coefficients</a>, Math. Comp., Vol. 20, No. 95 (1966), p. 465.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LogarithmicNumber.html">Logarithmic Number</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Gregory_coefficients">Gregory coefficients</a>.
%H Ming Wu and Hao Pan, <a href="http://www.fq.math.ca/Papers1/45-2/quartpan02_2007.pdf">Sums of products of Bernoulli numbers of the second kind</a>, Fib. Quart., Vol. 45, No. 2 (2007), pp. 146-150.
%H <a href="/index/Lo#logarithmic">Index entries for sequences related to logarithmic numbers</a>
%F 1/log(1+x) = Sum_{n>=-1} (a(n)/A002207(n)) * x^n. [corrected by _Robert Israel_, Oct 22 2015]
%F 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)).
%F 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
%F a(n)/A002207(n) = 1/(n+1)! * Sum_{k=0..n+1} Stirling1(n+1,k)/(k+1). - _Vladimir Kruchinin_, Sep 23 2012
%F G(n) = (Integral_{x=0..1} x*(x-n)_n)/(n+1)!, where (a)_n is the Pochhammer symbol. - _Vladimir Reshetnikov_, Oct 22 2015
%F 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
%F 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
%e 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
%p series(1/log(1+x),x,25);
%p with(combinat,stirling1):seq(numer(1/i!*sum(bernoulli(j)/(j)*stirling1(i,j-1), j=1..i+1)), i=1..24);
%t 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_ *)
%t Numerator@Table[Integrate[x Pochhammer[x - n, n], {x, 0, 1}]/(n + 1)!, {n, -1, 20}] (* _Vladimir Reshetnikov_, Oct 22 2015 *)
%o (Maxima) a(n):=sum(stirling1(n+1,k)/((n+1)!*(k+1)),k,0,n+1);
%o makelist(num(a(n)),n,-1,10); /* _Vladimir Kruchinin_, Sep 23 2012 */
%o (Maxima)
%o 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 */
%o (PARI) a(n) = numerator(sum(k=0, n+1, stirling(n+1, k, 1)/((n+1)!*(k+1)))); \\ _Michel Marcus_, Mar 20 2018
%o (Python)
%o from math import factorial
%o from fractions import Fraction
%o from sympy.functions.combinatorial.numbers import stirling
%o 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
%o (SageMath)
%o from functools import cache
%o @cache
%o def h(n):
%o return (-sum((-1)**k * h(n - k) / (k + 1) for k in range(1, n + 1))
%o + (-1)**n * n / (2*(n + 1)*(n + 2)))
%o def a(n): return h(n).numer() if n > 0 else 1
%o print([a(n) for n in range(-1, 20)]). # _Peter Luschny_, Dec 12 2023
%Y Cf. A001620, A002207, A006232, A006233, A002208, A002209, A002657, A002790.
%K sign,frac,nice
%O -1,5
%A _N. J. A. Sloane_
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