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A002675 Numerators of coefficients for central differences M_{4}^(2*n).
(Formerly M5035 N2173)
10
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 (list; graph; refs; listen; history; text; internal format)
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
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.
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.
Sequence in context: A146792 A124884 A052006 * A333855 A321217 A095748
KEYWORD
nonn,frac
AUTHOR
EXTENSIONS
More terms from Sean A. Irvine, Dec 20 2016
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)