login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A002672 Denominators of central difference coefficients M_{3}^(2n+1).
(Formerly M4588 N1957)
10
1, 8, 1920, 193536, 154828800, 1167851520, 892705701888000, 1428329123020800, 768472460034048000, 4058540589291090739200, 196433364521688791777280000, 5957759187690780937420800000, 30447485794244997427545243648000000, 341011840895543971188506728857600000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

From Peter Bala, Oct 03 2019: (Start)

Denominators in the expansion of (2*sinh(x/2))^3 = x^3 + (1/8)*x^5 + (13/1920)*x^7 + (41/193536)*x^9 + ....

Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^3 leads to a formula for the third central differences: f(x+3/2) - 3*f(x+1/2) + 3*f(x-1/2) - f(x-3/2) = (2*sinh(D/2))^3(f(x)) = D^3(f(x)) + (1/8)*D^5(f(x)) + (13/1920)* D^7(f(x)) + ..., where D denotes the differential operator d/dx. (End)

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

Table of n, a(n) for n=1..14.

H. E. Salzer, Tables of coefficients for obtaining central differences from the derivatives, Journal of Mathematics and Physics (this journal is also called Studies in Applied Mathematics), 42 (1963), 162-165, plus several inserted tables.

H. E. Salzer, Annotated scanned copy of left side of Table I.

E. W. Weisstein, Central Difference. From MathWorld--A Wolfram Web Resource.

FORMULA

a(n) = denominator(3! * m(3, 2 * n + 1) / (2 * n + 1)!) where m(k, k) = 1; m(k, q) = 0 for k = 0, k > q, or k + q odd; m(1, q) = 1/2^(q-1) for odd q; m(2, q) = 1 for even q; m(k, q+2) = m(k-2, q) + (k/2)^2 * m(k, q) otherwise. [From Salzer] - Sean A. Irvine, Dec 20 2016

CROSSREFS

Cf. A002673 (for numerators). Cf. A002671, A002674, A002675, A002676, A002677.

Sequence in context: A070903 A159386 A160302 * A027515 A297822 A297992

Adjacent sequences:  A002669 A002670 A002671 * A002673 A002674 A002675

KEYWORD

nonn,frac

AUTHOR

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified September 22 17:39 EDT 2021. Contains 347607 sequences. (Running on oeis4.)