|
%I M2017 N0797 #104 Dec 12 2023 08:41:00
%S 1,2,12,24,720,160,60480,24192,3628800,1036800,479001600,788480,
%T 2615348736000,475517952000,31384184832000,689762304000,
%U 32011868528640000,15613165568,786014494949376000,109285437800448000
%N Denominators of logarithmic numbers (also of Gregory coefficients G(n)).
%C Denominator of the determinant of the (n+1) X (n+1) matrix with 1's along the superdiagonal, (1/2)'s along the main diagonal, (1/3)'s along the subdiagonal, etc., and 0's everywhere else. - _John M. Campbell_, Dec 01 2011
%D 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
%D Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 266.
%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="/A002207/b002207.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, pp 133-149.
%H M. Alabdulmohsin, <a href="http://arxiv.org/abs/1209.5739">Summability Calculus</a>, arXiv:1209.5739v1 [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.
%H M. Coffey and J. 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., 121 (2012), 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., 27(1-2) (1924), 142-144.
%H A. N. Lowan and H. Salzer, <a href="http://dx.doi.org/10.1002/sapm194322149">Table of coefficients in numerical integration formulas</a>, J. Math. Phys., 22 (1943), 49-50.
%H A. N. Lowan and H. Salzer, <a href="/A002206/a002206.pdf">Table of coefficients in numerical integration formulas</a>, J. Math. Phys. Mass. Inst. Tech. 22 (1943), 49-50.[Annotated scanned copy]
%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://www.jstor.org/stable/2316623">Gregory's method for numerical integration</a>, Amer. Math. Monthly, 79 (1972), 270-274.
%H H. E. Salzer, <a href="http://dx.doi.org/10.1080/14786444708521604">Table of coefficients for repeated integration with differences</a>, Phil. Mag., 38 (1947), 331-336.
%H H. E. Salzer, <a href="/A002206/a002206_1.pdf">Table of coefficients for repeated integration with differences</a>, Phil. Mag., 38 (1947), 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 P. C. Stamper, <a href="http://dx.doi.org/10.1090/S0025-5718-66-99917-0">Table of Gregory coefficients</a>, Math. Comp., 20 (1966), 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., 45 (2007), 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} (A002206(n)/a(n)) * x^n.
%F 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
%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 A002206(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) = (1/(n+1)!)*Integral_{x=0..1} x*(x-n)_n, where (a)_n is the Pochhammer symbol. - _Vladimir Reshetnikov_, Oct 22 2015
%F a(n) = denominator(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(denom(1/i!*sum(bernoulli(j)/(j)*stirling1(i,j-1), j=1..i+1)), i=1..24);
%t Table[Denominator[Det[Array[Sum[KroneckerDelta[#1, #2+q]*1/(q+2)^1, {q, -1, n+1}] &, {n+1, n+1}]]], {n, 0, 20}] (* _John M. Campbell_, Dec 01 2011 *)
%t a[n_] := Denominator[n!^-1*Sum[BernoulliB[j]/j*StirlingS1[n, j-1], {j, 1, n+1}]]; a[-1] = 1; Table[a[n], {n, -1, 18}] (* _Jean-François Alcover_, May 16 2012, after Maple *)
%t Denominator@Table[Integrate[x Pochhammer[x - n, n], {x, 0, 1}]/(n + 1)!, {n, -1, 20}] (* _Vladimir Reshetnikov_, Oct 22 2015 *)
%o (PARI) a(n) = denominator(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 A002207(n): return (sum(Fraction(stirling(n+1,k,kind=1,signed=True),k+1) for k in range(n+2))/factorial(n+1)).denominator # _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).denom() if n > 0 else n + 2
%o print([a(n) for n in range(-1, 19)]). # _Peter Luschny_, Dec 12 2023
%Y Cf. A002206, A006232, A006233, A002208, A002209, A002657, A002790, A195189, A270857, A270859.
%K nonn,frac,nice
%O -1,2
%A _N. J. A. Sloane_
|
