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%I A002554 M4034 N1676 #41 Oct 15 2023 09:27:53
%S A002554 1,-5,259,-3229,117469,-7156487,2430898831,-60997921,141433003757,
%T A002554 -25587296781661,51270597630767,-6791120985104747,3400039831130408821,
%U A002554 -15317460638921852507,25789165074168004597399,-1550286106708510672406629,24823277118070193095631689
%N A002554 Numerators of coefficients for numerical differentiation.
%D A002554 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002554 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002554 Ruperto Corso, <a href="/A002554/b002554.txt">Table of n, a(n) for n = 1..387</a>
%H A002554 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 A002554 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 A002554 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.
%F A002554 a(n) is the numerator of (-1)^(n-1)*Cn-1{1^2..(2n-1)^2}/((2n)!*2^(2n-3)), where Cn{1^2..(2n+1)^2} equals 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. - _Ruperto Corso_, Dec 15 2011
%F A002554 a(n) = numerator(A001824(n-1)*(-1)^(n-1)/(2^(2*n-3)*(2*n)!)). - _Sean A. Irvine_, Mar 29 2014
%p A002554 with(combinat):
%p A002554 a:=n->add(mul(k, k=j), j=choose([seq((2*i-1)^2, i=1..n)], n-1))*(-1)^(n-1)/(2^(2*n-3)*(2*n)!):
%p A002554 seq(numer(a(n)), n=1..20); # _Ruperto Corso_, Dec 15 2011
%Y A002554 Cf. A001824, A002555.
%K A002554 sign,frac
%O A002554 1,2
%A A002554 _N. J. A. Sloane_
%E A002554 Corrected and extended by _Ruperto Corso_, Dec 15 2011

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