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%I M3145 N1274 #43 Oct 03 2021 22:18:56
%S 1,-1,-1,-3,-38,-135,-4315,-48125,-950684,-7217406,-682590930,
%T -6554931075,-903921420138,-10496162430897,-132415122967127,
%U -3606726811032345,-896549281211592008,-14008671728814262500,-4425739007479443851340
%N Coefficients for step-by-step integration.
%C All the terms except the first term are negative. - _Sean A. Irvine_, Nov 10 2013
%C a(n) / A002397(n) is the coefficient of the n-th forward difference of f in the estimate of y(x0) - y(x1).
%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 Jack W Grahl, <a href="/A002405/b002405.txt">Table of n, a(n) for n = 0..100</a>
%H Jack W Grahl, <a href="/A002405/a002405.pdf">Explanation of how the sequence was calculated</a>
%H Jack W Grahl, <a href="/A002405/a002405.py.txt">Python code to calculate this and related sequences</a>
%H W. F. Pickard, <a href="http://dx.doi.org/10.1145/321217.321226">Tables for the step-by-step integration of ordinary differential equations of the first order</a>, J. ACM 11 (1964), 229-233.
%H W. F. Pickard, <a href="/A002397/a002397.pdf">Tables for the step-by-step integration of ordinary differential equations of the first order</a>, J. ACM 11 (1964), 229-233. [Annotated scanned copy]
%F a(n) = lcm{1,2,...,n+1} * Sum_{k=0..n}((-1)^(n-k)/n+1-k)*s(-(n-1),k,n) where s(l,m,n) are the generalized Stirling numbers of the first kind. - _Sean A. Irvine_, Nov 10 2013
%Y With different signs, this is the leading diagonal of A260781.
%Y The coefficients used in numerical integration are given by fractions with A002397 as the denominators.
%Y A002401 is the corresponding sequence for the symmetric method of estimation.
%Y The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.
%K sign
%O 0,4
%A _N. J. A. Sloane_
%E More terms from _Sean A. Irvine_, Nov 10 2013
%E More terms from _Jack W Grahl_, Feb 28 2021
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