A002676 Denominators of coefficients for central differences M_{4}^(2*n). (Formerly M4282 N1789)

1, 6, 80, 30240, 1814400, 2661120, 871782912000, 3138418483200, 84687482880000, 170303140572364800, 1124000727777607680000, 724146127139635200000, 12703681025488077520896000000, 76222086152928465125376000000, 1531041037877004667453440000000

Offset

2,2

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 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)

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=2..16.

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 II.

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

Formula

a(n) = denominator(4! * m(4, 2 * n) / (2 * n)!) where m(k, q) is defined in A002672. - Sean A. Irvine, Dec 20 2016

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

Crossrefs

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

Sequence in context: A077393 A264694 A358037 * A052348 A309902 A196909

Adjacent sequences: A002673 A002674 A002675 * A002677 A002678 A002679

Keyword

nonn,frac

Author

N. J. A. Sloane

Extensions

More terms from Sean A. Irvine, Dec 20 2016

Status

approved