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A002405

Coefficients for step-by-step integration.

1, -1, -1, -3, -38, -135, -4315, -48125, -950684, -7217406, -682590930, -6554931075, -903921420138, -10496162430897, -132415122967127, -3606726811032345, -896549281211592008, -14008671728814262500, -4425739007479443851340

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,4

COMMENTS

All the terms except the first term are negative. - Sean A. Irvine, Nov 10 2013

a(n) / A002397(n) is the coefficient of the n-th forward difference of f in the estimate of y(x0) - y(x1).

REFERENCES

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

Jack W Grahl, Table of n, a(n) for n = 0..100

Jack W Grahl, Explanation of how the sequence was calculated

Jack W Grahl, Python code to calculate this and related sequences

W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233.

W. F. Pickard, Tables for the step-by-step integration of ordinary differential equations of the first order, J. ACM 11 (1964), 229-233. [Annotated scanned copy]

FORMULA

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

CROSSREFS

With different signs, this is the leading diagonal of A260781.

The coefficients used in numerical integration are given by fractions with A002397 as the denominators.

A002401 is the corresponding sequence for the symmetric method of estimation.