A002198 Denominators of coefficients for numerical integration.

24, 5760, 967680, 464486400, 122624409600, 2678117105664000, 64274810535936000, 149852129706639360000, 669659197233029971968000, 8839501403475995629977600000, 4879404774718749587747635200000

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

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.

Crossrefs

Cf. A002197.

See A000367, A006954, A008956 and A002671 for underlying sequences.

Factor of the LS1[ -2,n] matrix coefficients in A160487.

Programs

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

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

Extensions

Two more terms and editing by Johannes W. Meijer, Sep 21 2012