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A002206
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Numerators of logarithmic numbers (also of Gregory coefficients G(n)).
(Formerly M5066 N2194)
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14
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1, 1, -1, 1, -19, 3, -863, 275, -33953, 8183, -3250433, 4671, -13695779093, 2224234463, -132282840127, 2639651053, -111956703448001, 50188465, -2334028946344463, 301124035185049, -12365722323469980029
(list; graph; refs; listen; history; internal format)
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OFFSET
| -1,5
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COMMENTS
| For n>0 G(n)=(-1)^(n+1)*int(1/[(ln^2(x)+Pi^2)*(x+1)^n],x=0..infinity). G(1)=1/2 for n>1 G(n)=(-1)^(n+1)/(n+1)-sum((-1)^k*G(n-k)/(k+1),k=1..n-1). Euler constant is given by : gamma=sum((-1)^(n+1)*G(n)/n,n=1..infinity) [From Groux Roland (roland.groux(AT)orange.fr), Jan 14 2009]
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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.
M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990, see page 101 [Problem 87-6].
A. N. Lowan and H. Salzer, Table of coefficients in numerical integration formulae, J. Math. Phys. Mass. Inst. Tech. 22 (1943), 49-50.
H. E. Salzer, Table of coefficients for repeated integration with differences, Phil. Mag., 38 (1947), 331-336.
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).
P. C. Stamper, Table of Gregory coefficients, Math. Comp., 20 (1966), 465.
Ming Wu and Hao Pan, Sums of products of Bernoulli numbers of the second kind, Fib. Quart., 45 (2007), 146-150.
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LINKS
| T. D. Noe, Table of n, a(n) for n=-1..100
G. M. Phillips, Gregory's method for numerical integration, Amer. Math. Monthly, 79 (1972), 270-274.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Index entries for sequences related to logarithmic numbers
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FORMULA
| G.f.: 1/log(1+x).
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)=A002206(n)/A002207=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
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EXAMPLE
| 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
G(0), G(1), ... = 0, 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
| 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);
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CROSSREFS
| Cf. A002207, A006232, A006233, A002208, A002209, A002657, A002790.
Sequence in context: A040353 A128160 A092120 * A040349 A040350 A089572
Adjacent sequences: A002203 A002204 A002205 * A002207 A002208 A002209
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KEYWORD
| sign,frac,nice
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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