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

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

Offset: 2

Author

N. J. A. Sloane

Keywords

• nonn
• frac

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

• 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.

Crossrefs

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

Programs

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

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

Extensions

More terms from Sean A. Irvine, Dec 20 2016