A002405 Coefficients for step-by-step integration. (Formerly M3145 N1274)

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

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.

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

Sequence in context: A093939 A129122 A345688 * A050394 A281798 A156071

Adjacent sequences: A002402 A002403 A002404 * A002406 A002407 A002408

Keyword

sign

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Nov 10 2013

More terms from Jack W Grahl, Feb 28 2021

Status

approved