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Search: id:A002208
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| A002208 |
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Numerators of coefficients for numerical integration.
(Formerly M3737 N1527)
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+0
9
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| 1, 1, 5, 3, 251, 95, 19087, 5257, 1070017, 25713, 26842253, 4777223, 703604254357, 106364763817, 1166309819657, 25221445, 8092989203533249, 85455477715379, 12600467236042756559, 1311546499957236437, 8136836498467582599787
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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a(n) is (-1)^n times the numerator of the "reverse" multiple zeta value zeta_n^R(0,0,...,0) for n>0. - Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Nov 29 2006
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
S. Akiyama and Y. Tanigawa, Multiple zeta values at non-positive integers, Ramanujan J. 5 (2001), 327-351.
E. Isaacson and H. B. Keller, Analysis of Numerical Methods, ISBN 0 471 42865 5, 1966, John Wiley and Sons, pp. 318-319.
Charles Jordan, Calculus of Finite Differences, Chelsea 1965, p. 529.
A. N. Lowan and H. Salzer, Table of coefficients in numerical integration formulae, J. Math. Phys. Mass. Inst. Tech. 22 (1943), 49-50.
Guodong Liu, Some computational formulas for Norlund numbers, Fib. Quart., 45 (2007), 133-137.
N. E. Noerlund, Vorlesungen ueber Differenzenrechnung, Springer-Verlag, Berlin, 1924.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..100
Index entries for sequences related to Bernoulli numbers.
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FORMULA
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G.f.: -x/((1-x)*ln(1-x)).
(From Rudi Huysmans, rudi_huysmans(AT)hotmail.com) Let K_i = A002208(i)/A002209(i), with K_1=1/2; K_2=-5/12; ... and [i n] = Stirling numbers of the first kind (e.g.[4 2] = 11), {i n} = Stirling numbers of the second kind (e.g. {4 2}=7) and B_i the Bernoulli numbers. Then K_i =((-1)^i / (i-1)! ).Sum_n=1..i [i n].B_n/n and B_i = i.(-1)^i. Sum_n=1..i {i n}.(n-1)!.K_n.
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EXAMPLE
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1, 1/2, 5/12, 3/8, 251/720, 95/288, 19087/60480, 5257/17280, 1070017/3628800, 25713/89600, 26842253/95800320, 4777223/17418240, 703604254357/2615348736000, 106364763817/402361344000, ... = A002208/A002209
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CROSSREFS
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Cf. A002209. See also A002657, A002790, A006232, A006233, A002206, A002207.
Sequence in context: A109254 A145985 A048885 this_sequence A100653 A121021 A159799
Adjacent sequences: A002205 A002206 A002207 this_sequence A002209 A002210 A002211
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KEYWORD
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frac,nonn,easy,nice
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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