%I M5035 N2173 #34 Oct 06 2019 06:51:08

%S 1,1,1,17,31,1,5461,257,73,1271,60787,241,22369621,617093,49981,

%T 16843009,5726623061,7957,91625968981,61681,231927781,50991843607,

%U 499069107643,4043309297,1100586419201,5664905191661,1672180312771

%N Numerators of coefficients for central differences M_{4}^(2*n).

%C From _Peter Bala_, Oct 03 2019: (Start)

%C Numerators in the expansion of (2*sinh(x/2))^4 = x^4 + (1/6)*x^6 + (1/80)*x^8 + (17/30240)*x^10 + ....

%C 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)

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H 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 H. E. Salzer, <a href="/A002675/a002675.png">Annotated scanned copy of left side of Table II</a>.

%H H. E. Salzer, <a href="/A002677/a002677.png">Annotated scanned copy of left side of Table III</a>

%H E. W. Weisstein, <a href="http://mathworld.wolfram.com/CentralDifference.html">Central Difference</a>. From MathWorld--A Wolfram Web Resource.

%p gf := (sinh(2*sqrt(x)) - 2*sinh(sqrt(x)))*sqrt(x):

%p ser := series(gf, x, 40): seq(numer(coeff(ser,x,n)), n=2..28); # _Peter Luschny_, Oct 05 2019

%Y Cf. A002676 and A002677 (two different choices for denominators).

%Y Also equals A002430/A002431.

%Y Cf. A002671, A002672, A002673, A002674.

%K nonn,frac

%O 2,4

%A _N. J. A. Sloane_

%E More terms from _Sean A. Irvine_, Dec 20 2016