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%I A002553 M5166 N2243 #31 Jul 18 2015 13:17:00
%S A002553 1,24,640,7168,294912,2883584,54525952,167772160,36507222016,
%T A002553 326417514496,5772436045824,50577534877696,1759218604441600,
%U A002553 15199648742375424,261208778387488768,2233785415175766016,101457092405402533888
%N A002553 Coefficients for numerical differentiation.
%D A002553 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002553 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002553 W. G. Bickley and J. C. P. Miller, <a href="http://dx.doi.org/10.1080/14786444208521334">Numerical differentiation near the limits of a difference table</a>, Phil. Mag., 33 (1942), 1-12 (plus tables).
%H A002553 W. G. Bickley and J. C. P. Miller, <a href="/A002551/a002551.pdf">Numerical differentiation near the limits of a difference table</a>, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]
%H A002553 T. R. Van Oppolzer, <a href="http://www.archive.org/stream/lehrbuchzurbahnb02oppo#page/23/mode/1up">Lehrbuch zur Bahnbestimmung der Kometen und Planeten</a>, Vol. 2, Engelmann, Leipzig, 1880, p. 23, (see denominators of numbers named M(1,2k+1)).
%F A002553 a(n) = denom(A001818(n)*(-1)^(n-1)/(2^(2*n)*(2*n+1)!)). - _Sean A. Irvine_, Mar 29 2014
%F A002553 a(n) is the denominator of(-1)^(n-1)*Cn-1{1^2..(2n-1)^2}/((2n+1)!*2^(2n)), where Cn{1^2..(2n+1)^2} is equal to 1 when n=0, otherwise, it is the sum of the products of all possible combinations, of size n, of the numbers (2k+1)^2 with k=0,1,..,n. - _Sean A. Irvine_, after _Ruperto Corso_, Mar 29 2014
%p A002553 with(combinat): a:=n->add(mul(k, k=j), j=choose([seq((2*i-1)^2, i=1..n)], n))*(-1)^(n-1)/(2^(2*n)*(2*n+1)!):seq(a(n), n=0..20); # _Sean A. Irvine_, after _Ruperto Corso_
%Y A002553 Cf. A001818, A002555.
%K A002553 nonn
%O A002553 0,2
%A A002553 _N. J. A. Sloane_
%E A002553 More terms from _Sean A. Irvine_, Mar 29 2014

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