OFFSET
0,1
COMMENTS
The denominators of these coefficients for numerical integration are a combination of the Bernoulli numbers B{2k}, the central factorial numbers A008956(n, k) and the factor 4^n*(2*n+1)!. - Johannes W. Meijer, Jan 27 2009
REFERENCES
H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218.
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
H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218. [Annotated scanned copy]
T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 545.
FORMULA
a(n) = denominator(Sum_{k=1..n+1}((1-2^(2*k-1))*(-1)^k*(B_{2k}/(2*k))*A008956(n, n+1-k)) / (2*4^(n)*(2*n+1)!)) for n >= 0. - Johannes W. Meijer, Jan 27 2009
MAPLE
nmax:=10: for n from 0 to nmax do A008956(n, 0) := 1: A008956(n, n) := (doublefactorial(2*n-1))^2 od: for n from 1 to nmax do for k from 1 to n-1 do A008956(n, k) := (2*n-1)^2*A008956(n-1, k-1) + A008956(n-1, k) od: od:
for n from 0 to nmax do Delta(n) := add((1-2^(2*k1-1)) * (-1)^k1 * (bernoulli(2*k1)/(2*k1)) * A008956(n, n+1-k1), k1=1..n+1) / (2*4^(n)*(2*n+1)!) end do: a:=n-> denom (Delta(n)): seq(a(n), n=0..nmax); # Johannes W. Meijer, Jan 27 2009, Revised Sep 21 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
EXTENSIONS
Two more terms and editing by Johannes W. Meijer, Sep 21 2012
STATUS
approved