The On-Line Encyclopedia of Integer Sequences (OEIS)

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A002676

Denominators of coefficients for central differences M_{4}^(2*n).
(history; published version)

#32 by Charles R Greathouse IV at Sun Feb 16 08:32:26 EST 2025

LINKS

E. W. Weisstein, <a href="httphttps://mathworld.wolfram.com/CentralDifference.html">Central Difference</a>. From MathWorld--A Wolfram Web Resource.

Discussion

Sun Feb 16

08:32

OEIS Server: https://oeis.org/edit/global/3014

#31 by Susanna Cuyler at Sat Oct 05 09:14:05 EDT 2019

STATUS

reviewed

approved

#30 by Joerg Arndt at Sat Oct 05 07:54:55 EDT 2019

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proposed

reviewed

#29 by Peter Luschny at Sat Oct 05 07:33:02 EDT 2019

STATUS

editing

proposed

#28 by Peter Luschny at Sat Oct 05 07:32:56 EDT 2019

MAPLE

gf := 6 - 8*cosh(sqrt(x)) + 2*cosh(2*sqrt(x)): ser := series(gf, x, 40):

seq(denom(coeff(ser, x, n)), n=2..16); # Peter Luschny, Oct 05 2019

STATUS

approved

editing

#27 by Michel Marcus at Fri Oct 04 09:21:28 EDT 2019

STATUS

reviewed

approved

#26 by Joerg Arndt at Fri Oct 04 09:07:08 EDT 2019

STATUS

proposed

reviewed

#25 by Peter Bala at Fri Oct 04 07:46:33 EDT 2019

STATUS

editing

proposed

#24 by Peter Bala at Thu Oct 03 12:26:26 EDT 2019

COMMENTS

Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^4 leads to a formula for the fourth central differences: f(x+2) - 4*f(x+1) + 6*f(x) - 4*f(x-1) + f(x-2) = (2*sinh(D/2))^4(f(x)) = D^4(f(x)) + (1/6)*D^6(f(x)) + (1/80)*D^8(f(x)) + (17/30240)*D^10(f(x)) + ..., where D denotes the differential operator d/dx. (End)

LINKS

E. W. Weisstein, <a href="http://mathworld.wolfram.com/CentralDifference.html">Central Difference</a>. From MathWorld--A Wolfram Web Resource.

#23 by Peter Bala at Thu Oct 03 12:18:36 EDT 2019

COMMENTS

From Peter Bala, Oct 03 2019: (Start)

Denominators in the expansion of (2*sinh(x/2))^4 = x^4 + (1/6)*x^6 + (1/80)*x^8 + (17/30240)*x^10 + ....

Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^4 leads to a formula for the central differences: f(x+2) - 4*f(x+1) + 6*f(x) - 4*f(x-1) + f(x-2) = (2*sinh(D/2))^4(f(x)) = D^4(f(x)) + (1/6)*D^6(f(x)) + (1/80)*D^8(f(x)) + (17/30240)*D^10(f(x)) + ..., where D denotes the differential operator d/dx. (End)

CROSSREFS

Cf. A002675 (numerators). Cf. A002671, A002672, A002673, A002674, A002677.

STATUS

approved

editing