A002675 - OEIS

A002675

Numerators of coefficients for central differences M_{4}^(2*n).
(Formerly M5035 N2173)

1, 1, 1, 17, 31, 1, 5461, 257, 73, 1271, 60787, 241, 22369621, 617093, 49981, 16843009, 5726623061, 7957, 91625968981, 61681, 231927781, 50991843607, 499069107643, 4043309297, 1100586419201, 5664905191661, 1672180312771

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OFFSET

2,4

COMMENTS

From Peter Bala, Oct 03 2019: (Start)

Numerators 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..28.

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.

H. E. Salzer, Annotated scanned copy of left side of Table III

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

MAPLE

gf := (sinh(2*sqrt(x)) - 2*sinh(sqrt(x)))*sqrt(x):

ser := series(gf, x, 40): seq(numer(coeff(ser, x, n)), n=2..28); # Peter Luschny, Oct 05 2019

CROSSREFS

Cf. A002676 and A002677 (two different choices for denominators).

Also equals A002430/A002431.

Cf. A002671, A002672, A002673, A002674.

Sequence in context: A146792 A124884 A052006 * A333855 A321217 A095748

Adjacent sequences: A002672 A002673 A002674 * A002676 A002677 A002678