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A002672 Denominators of central difference coefficients M_{3}^(2n+1).

%I A002672 M4588 N1957 #31 Feb 16 2025 08:32:26
%S A002672 1,8,1920,193536,154828800,1167851520,892705701888000,
%T A002672 1428329123020800,768472460034048000,4058540589291090739200,
%U A002672 196433364521688791777280000,5957759187690780937420800000,30447485794244997427545243648000000,341011840895543971188506728857600000
%N A002672 Denominators of central difference coefficients M_{3}^(2n+1).
%C A002672 From _Peter Bala_, Oct 03 2019: (Start)
%C A002672 Denominators in the expansion of (2*sinh(x/2))^3 = x^3 + (1/8)*x^5 + (13/1920)*x^7 + (41/193536)*x^9 + ....
%C A002672 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)
%D A002672 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002672 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002672 H. E. Salzer, <a href="https://doi.org/10.1002/sapm1963421162">Tables of coefficients for obtaining central differences from the derivatives</a>, Journal of Mathematics and Physics (this journal is also called Studies in Applied Mathematics), 42 (1963), 162-165, plus several inserted tables.
%H A002672 H. E. Salzer, <a href="/A002673/a002673.png">Annotated scanned copy of left side of Table I</a>.
%H A002672 E. W. Weisstein, <a href="https://mathworld.wolfram.com/CentralDifference.html">Central Difference</a>. From MathWorld--A Wolfram Web Resource.
%F A002672 a(n) = denominator(3! * m(3, 2 * n + 1) / (2 * n + 1)!) where m(k, k) = 1; m(k, q) = 0 for k = 0, k > q, or k + q odd; m(1, q) = 1/2^(q-1) for odd q; m(2, q) = 1 for even q; m(k, q+2) = m(k-2, q) + (k/2)^2 * m(k, q) otherwise. [From Salzer] - _Sean A. Irvine_, Dec 20 2016
%Y A002672 Cf. A002673 (for numerators). Cf. A002671, A002674, A002675, A002676, A002677.
%K A002672 nonn,frac
%O A002672 1,2
%A A002672 _N. J. A. Sloane_