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A002207
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Denominators of logarithmic numbers (also of Gregory coefficients G(n)).
(Formerly M2017 N0797)
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25
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1, 2, 12, 24, 720, 160, 60480, 24192, 3628800, 1036800, 479001600, 788480, 2615348736000, 475517952000, 31384184832000, 689762304000, 32011868528640000, 15613165568, 786014494949376000, 109285437800448000
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OFFSET
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-1,2
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COMMENTS
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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
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REFERENCES
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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).
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LINKS
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FORMULA
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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)).
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
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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(denom(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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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 *)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn,frac,nice
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AUTHOR
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STATUS
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approved
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