A002554 Numerators of coefficients for numerical differentiation.

1, -5, 259, -3229, 117469, -7156487, 2430898831, -60997921, 141433003757, -25587296781661, 51270597630767, -6791120985104747, 3400039831130408821, -15317460638921852507, 25789165074168004597399, -1550286106708510672406629, 24823277118070193095631689

Offset: 1

Author

N. J. A. Sloane

Keywords

• sign
• frac

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

• Ruperto Corso, Table of n, a(n) for n = 1..387
• W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables).
• W. G. Bickley and J. C. P. Miller, Numerical differentiation near the limits of a difference table, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]
• T. R. Van Oppolzer, Lehrbuch zur Bahnbestimmung der Kometen und Planeten, Vol. 2, Engelmann, Leipzig, 1880, p. 23.

Crossrefs

Cf. A001824, A002555.

Programs

Maple

with(combinat):
a:=n->add(mul(k, k=j), j=choose([seq((2*i-1)^2, i=1..n)], n-1))*(-1)^(n-1)/(2^(2*n-3)*(2*n)!):
seq(numer(a(n)), n=1..20); # Ruperto Corso, Dec 15 2011

Formula

a(n) is the numerator of (-1)^(n-1)*Cn-1{1^2..(2n-1)^2}/((2n)!*2^(2n-3)), where Cn{1^2..(2n+1)^2} equals 1 when n=0, otherwise it is the sum of the products of all possible combinations, of size n, of the numbers (2k+1)^2 with k=0,1,...,n. - Ruperto Corso, Dec 15 2011

a(n) = numerator(A001824(n-1)*(-1)^(n-1)/(2^(2*n-3)*(2*n)!)). - Sean A. Irvine, Mar 29 2014

Extensions

Corrected and extended by Ruperto Corso, Dec 15 2011