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 A002673 Numerators of central difference coefficients M_{3}^(2n+1). (Formerly M4894 N2097) 9
 1, 1, 13, 41, 671, 73, 597871, 7913, 28009, 792451, 170549237, 19397633, 317733228541, 9860686403, 75397891, 170314355593, 2084647712458321, 29327731093, 168856464709124011, 3063310184201, 499338236699611, 535201577273701757, 23571643935246013553 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS From Peter Bala, Oct 03 2019: (Start) Numerators in the expansion of (2*sinh(x/2))^3 = x^3 + (1/8)*x^5 + (13/1920)*x^7 + (41/193536)*x^9 + .... Let f(x) be a polynomial in x. The expansion of (2*sinh(x/2))^3 leads to a formula for the third central differences: f(x+3/2) - 3*f(x+1/2) + 3*f(x-1/2) - f(x-3/2) = (2*sinh(D/2))^3(f(x)) = D^3(f(x)) + (1/8)*D^5(f(x)) + (13/1920)* D^7(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. H. E. Salzer, Annotated scanned copy of left side of Table I. E. W. Weisstein, Central Difference. From MathWorld--A Wolfram Web Resource. CROSSREFS Cf. A002671, A002672, A002674, A002675, A002676, A002677. Sequence in context: A201121 A116153 A116209 * A081301 A159527 A139277 Adjacent sequences:  A002670 A002671 A002672 * A002674 A002675 A002676 KEYWORD nonn,frac AUTHOR STATUS approved

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Last modified September 18 04:22 EDT 2021. Contains 347508 sequences. (Running on oeis4.)