A002401 Coefficients for step-by-step integration. (Formerly M3947 N1626)

1, 1, 5, 27, 502, 2375, 95435, 1287965, 29960476, 262426878, 28184365650, 303473091075, 46437880787562, 593196287807409, 8172332906336599, 241563260379065625, 64808657541894257992, 1087738506483388123364, 367580830209839294339148, 6906008426663826491899602, 136666305828261517346022452

Offset

0,3

Comments

These are the coefficients of the n-th forward difference of f in the estimate for y(x1) - y(x0), also the coefficients of f(x0) in the estimate for 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 this sequence is 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+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

Column 0 of A260781.

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: A360732 A135627 A244655 * A034946 A321072 A119941

Adjacent sequences: A002398 A002399 A002400 * A002402 A002403 A002404

Keyword

nonn

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