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A002677 Denominators of coefficients for central differences M_{3}'^(2*n+1).

%I A002677 M3676 N1499 #38 Oct 09 2019 07:21:25
%S A002677 1,4,40,12096,604800,760320,217945728000,697426329600,16937496576000,
%T A002677 30964207376793600,187333454629601280000,111407096483020800000,
%U A002677 1814811575069725360128000000,10162944820390462016716800000
%N A002677 Denominators of coefficients for central differences M_{3}'^(2*n+1).
%D A002677 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002677 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002677 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 A002677 H. E. Salzer, <a href="/A002677/a002677.png">Annotated scanned copy of left side of Table III</a>
%F A002677 From _Peter Bala_, Oct 03 2019: (Start)
%F A002677 a(n) are the denominators in the expansion of (1/2)*(d/dx)(2*sinh(sqrt(x)/2))^4 =
%F A002677 x + (1/4)*x^2 + (1/40)*x^3 + (17/12096)*x^4 + (31/604800)*x^5 + ...
%F A002677 The a(n) also appear as denominators in the difference formula: (1/2)*f(x+2) - f(x+1) + f(x-1) - (1/2)*f(x-2) = D^3(f(x)) + (1/4)*D^5(f(x)) + (1/40)*D^7(f(x)) + (17/12096)*D^9(f(x)) + ..., where D denotes the differential operator d/dx.
%F A002677 (End)
%p A002677 gf := (sinh(2*sqrt(x)) - 2*sinh(sqrt(x)))/sqrt(x): ser := series(gf, x, 20):
%p A002677 seq(denom(coeff(ser, x, n)), n=1..14); # _Peter Luschny_, Oct 05 2019
%Y A002677 Numerators are A002675.
%Y A002677 Cf. A002671, A002672, A002673, A002674, A002676.
%K A002677 nonn,frac
%O A002677 1,2
%A A002677 _N. J. A. Sloane_
%E A002677 More terms from _Sean A. Irvine_, Dec 20 2016