

A002676


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


6



1, 6, 80, 30240, 1814400, 2661120, 871782912000, 3138418483200, 84687482880000, 170303140572364800, 1124000727777607680000, 724146127139635200000, 12703681025488077520896000000, 76222086152928465125376000000, 1531041037877004667453440000000
(list;
graph;
refs;
listen;
history;
text;
internal format)



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(x1) + f(x2) = (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), 162165, plus several inserted tables.
H. E. Salzer, Annotated scanned copy of left side of Table II.
E. W. Weisstein, Central Difference. From MathWorldA 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: A323694 A077393 A264694 * 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



