The On-Line Encyclopedia of Integer Sequences (OEIS)

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A002554

Numerators of coefficients for numerical differentiation.
(history; published version)

#41 by Michael De Vlieger at Sun Oct 15 09:27:53 EDT 2023

STATUS

proposed

approved

#40 by Joerg Arndt at Sun Oct 15 03:51:51 EDT 2023

STATUS

editing

proposed

Discussion

Sun Oct 15

06:24

Stefano Spezia: Maple code is fine

#39 by Joerg Arndt at Sun Oct 15 03:51:24 EDT 2023

MAPLE

with(combinat):

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

seq(numer(a(n)), n=1..20); # Ruperto Corso, Dec 15 2011

STATUS

proposed

editing

Discussion

Sun Oct 15

03:51

Joerg Arndt: someone please check my reformatting of the Maple code

#38 by Jon E. Schoenfield at Sat Oct 14 16:14:31 EDT 2023

STATUS

editing

proposed

Discussion

Sun Oct 15

01:27

Michel Marcus: C_(n-1) where C_n{1^2..(2n+1)^2} equals 1 when n=0, otherwise it is ....

#37 by Jon E. Schoenfield at Sat Oct 14 16:10:18 EDT 2023

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

STATUS

approved

editing

Discussion

Sat Oct 14

16:14

Jon E. Schoenfield: In the Formula section, I don’t understand the notation at “(-1)^(n-1)*Cn-1{1^2..(2n-1)^2}” — the “1” in “Cn-1” isn’t just a coefficient, to be multiplied by what follows it … is it? Should “Cn-1” be “C_(n-1)”? Or something else?

#36 by Alois P. Heinz at Wed Feb 27 19:03:32 EST 2019

STATUS

proposed

approved

#35 by Michel Marcus at Wed Feb 27 12:02:18 EST 2019

STATUS

editing

proposed

#34 by Michel Marcus at Wed Feb 27 12:02:15 EST 2019

LINKS

T. R. Van Oppolzer, <a href="http://www.archive.org/stream/lehrbuchzurbahnb02oppo#page/23/mode/1up">Lehrbuch zur Bahnbestimmung der Kometen und Planeten</a>, Vol. 2, Engelmann, Leipzig, 1880, p. 23.

FORMULA

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

STATUS

approved

editing

#33 by N. J. A. Sloane at Sat Jul 18 13:17:27 EDT 2015

STATUS

editing

approved

#32 by N. J. A. Sloane at Sat Jul 18 13:17:24 EDT 2015

LINKS

W. G. Bickley and J. C. P. Miller, <a href="/A002551/a002551.pdf">Numerical differentiation near the limits of a difference table</a>, Phil. Mag., 33 (1942), 1-12 (plus tables) [Annotated scanned copy]

STATUS

approved

editing