|
| |
|
|
A002208
|
|
Numerators of coefficients for numerical integration.
(Formerly M3737 N1527)
|
|
11
|
|
|
|
1, 1, 5, 3, 251, 95, 19087, 5257, 1070017, 25713, 26842253, 4777223, 703604254357, 106364763817, 1166309819657, 25221445, 8092989203533249, 85455477715379, 12600467236042756559, 1311546499957236437, 8136836498467582599787
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
0,3
|
|
|
COMMENTS
|
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, Nov 29 2006
a(n) = T191578(2*n,n)/(2*n)!, n>0. [Vladimir Kruchinin, Feb 02 2013]
|
|
|
REFERENCES
|
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.
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 = 0..100
Index entries for sequences related to Bernoulli numbers .
|
|
|
FORMULA
|
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.
a(n) = numerator((-1)^n*sum(k=0..n, (k!*stirling2(n,k)* stirling1(n+k,n))/(n+k)!)). [Vladimir Kruchinin, Feb 02 2013]
|
|
|
EXAMPLE
|
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
|
|
|
MATHEMATICA
|
Numerator/@CoefficientList[Series[-x/((1-x)Log[1-x]), {x, 0, 20}], x] (* Harvey P. Dale, May 04 2011 *)
a[0] = 1; a[n_] := (-1)^n*Sum[(-1)^(k+1)*BernoulliB[k]*StirlingS1[n, k]/k, {k, 1, n}]/(n-1)!; Table[a[n], {n, 0, 20}] // Numerator (* Jean-François Alcover, Sep 27 2012, after Rudi Huysmans's formula *)
|
|
|
CROSSREFS
|
Cf. A002209. See also A002657, A002790, A006232, A006233, A002206, A002207.
Sequence in context: A145985 A180403 A048885 * A100653 A105318 A121021
Adjacent sequences: A002205 A002206 A002207 * A002209 A002210 A002211
|
|
|
KEYWORD
|
frac,nonn,easy,nice
|
|
|
AUTHOR
|
N. J. A. Sloane.
|
|
|
STATUS
|
approved
|
| |
|
|
