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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 (list; graph; refs; listen; history; text; internal format)



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)


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).


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.


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


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


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




N. J. A. Sloane


More terms from Sean A. Irvine, Dec 20 2016



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Last modified November 30 17:12 EST 2021. Contains 349424 sequences. (Running on oeis4.)