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

Offset: 0

Author

N. J. A. Sloane

Keywords

• sign

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]

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.

The following sequences are taken from page 231 of Pickard (1964): A002397, A002398, A002399, A002400, A002401, A002402, A002403, A002404, A002405, A002406, A260780, A260781.

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

Extensions

More terms from Sean A. Irvine, Nov 10 2013

More terms from Jack W Grahl, Feb 28 2021