OFFSET
-1,2
COMMENTS
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
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.
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, pp. 133-149.
Ibrahim M. Alabdulmohsin, Summability Calculus, arXiv:1209.5739v1 [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.
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.
A. N. Lowan and H. Salzer, Table of coefficients in numerical integration formulas, J. Math. Phys., 22 (1943), 49-50.
A. N. Lowan and H. Salzer, Table of coefficients in numerical integration formulas, J. Math. Phys. Mass. Inst. Tech. 22 (1943), 49-50.[Annotated scanned copy]
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, 79 (1972), 270-274.
H. E. Salzer, Table of coefficients for repeated integration with differences, Phil. Mag., 38 (1947), 331-336.
H. E. Salzer, Table of coefficients for repeated integration with differences, Phil. Mag., 38 (1947), 331-336. [Annotated scanned copy]
Raphael Schumacher, Rapidly Convergent Summation Formulas involving Stirling Series, arXiv preprint arXiv:1602.00336, 2016
P. C. Stamper, Table of Gregory coefficients, Math. Comp., 20 (1966), 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., 45 (2007), 146-150.
FORMULA
1/log(1+x) = Sum_{n>=-1} (A002206(n)/a(n)) * x^n.
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), Jan 21 2002
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+1)!)*Sum_{k=0..n+1} Stirling1(n+1,k)/(k+1). - Vladimir Kruchinin, Sep 23 2012
G(n) = (1/(n+1)!)*Integral_{x=0..1} x*(x-n)_n dx, where (a)_n is the Pochhammer symbol. - Vladimir Reshetnikov, Oct 22 2015
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
EXAMPLE
MAPLE
series(1/log(1+x), x, 25);
with(combinat, stirling1):seq(denom(1/i!*sum(bernoulli(j)/(j)*stirling1(i, j-1), j=1..i+1)), i=1..24);
MATHEMATICA
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 *)
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 *)
Denominator@Table[Integrate[x Pochhammer[x - n, n], {x, 0, 1}]/(n + 1)!, {n, -1, 20}] (* Vladimir Reshetnikov, Oct 22 2015 *)
Denominator@CoefficientList[x/Log[1+x] + O[x]^20, x] (* Oliver Seipel, Jul 06 2024 *)
PROG
(PARI) a(n) = denominator(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 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
(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).denom() if n > 0 else n + 2
print([a(n) for n in range(-1, 19)]). # Peter Luschny, Dec 12 2023
CROSSREFS
KEYWORD
nonn,frac,nice
AUTHOR
STATUS
approved