This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A002675 M5035 N2173 #35 Feb 16 2025 08:32:26
%S A002675 1,1,1,17,31,1,5461,257,73,1271,60787,241,22369621,617093,49981,
%T A002675 16843009,5726623061,7957,91625968981,61681,231927781,50991843607,
%U A002675 499069107643,4043309297,1100586419201,5664905191661,1672180312771
%N A002675 Numerators of coefficients for central differences M_{4}^(2*n).
%C A002675 From _Peter Bala_, Oct 03 2019: (Start)
%C A002675 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 A002675 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 A002675 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D A002675 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A002675 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 A002675 H. E. Salzer, <a href="/A002675/a002675.png">Annotated scanned copy of left side of Table II</a>.
%H A002675 H. E. Salzer, <a href="/A002677/a002677.png">Annotated scanned copy of left side of Table III</a>
%H A002675 E. W. Weisstein, <a href="https://mathworld.wolfram.com/CentralDifference.html">Central Difference</a>. From MathWorld--A Wolfram Web Resource.
%p A002675 gf := (sinh(2*sqrt(x)) - 2*sinh(sqrt(x)))*sqrt(x):
%p A002675 ser := series(gf, x, 40): seq(numer(coeff(ser,x,n)), n=2..28); # _Peter Luschny_, Oct 05 2019
%Y A002675 Cf. A002676 and A002677 (two different choices for denominators).
%Y A002675 Also equals A002430/A002431.
%Y A002675 Cf. A002671, A002672, A002673, A002674.
%K A002675 nonn,frac
%O A002675 2,4
%A A002675 _N. J. A. Sloane_
%E A002675 More terms from _Sean A. Irvine_, Dec 20 2016