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%I M5178 N2250 #31 Mar 13 2019 11:33:36
%S 24,5760,967680,464486400,122624409600,2678117105664000,
%T 64274810535936000,149852129706639360000,669659197233029971968000,
%U 8839501403475995629977600000,4879404774718749587747635200000
%N Denominators of coefficients for numerical integration.
%C 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
%D H. E. Salzer, Coefficients for mid-interval numerical integration with central differences, Phil. Mag., 36 (1945), 216-218.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H T. D. Noe, <a href="/A002198/b002198.txt">Table of n, a(n) for n = 0..100</a>
%H H. E. Salzer, <a href="/A002197/a002197.pdf">Coefficients for mid-interval numerical integration with central differences</a>, Phil. Mag., 36 (1945), 216-218. [Annotated scanned copy]
%H T. R. Van Oppolzer, <a href="http://www.archive.org/stream/lehrbuchzurbahnb02oppo#page/545/mode/1up">Lehrbuch zur Bahnbestimmung der Kometen und Planeten</a>, Vol. 2, Engelmann, Leipzig, 1880, p. 545.
%F 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
%p 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:
%p 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
%Y Cf. A002197.
%Y See A000367, A006954, A008956 and A002671 for underlying sequences.
%Y Factor of the LS1[ -2,n] matrix coefficients in A160487.
%K nonn
%O 0,1
%A _N. J. A. Sloane_
%E Two more terms and editing by _Johannes W. Meijer_, Sep 21 2012
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