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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 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 KEYWORD nonn,frac AUTHOR N. J. A. Sloane EXTENSIONS More terms from Sean A. Irvine, Dec 20 2016 STATUS approved

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Last modified November 29 07:03 EST 2023. Contains 367429 sequences. (Running on oeis4.)