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 A285044 #11 Nov 22 2024 00:35:15 %S A285044 1,50,550,4020,24710,138012,725340,3655080,17859270,85230860, %T A285044 399257716,1842353240,8396404380,37868584600,169278679800, %U A285044 750923914320,3308947546950,14495583969900,63172016823300,274031830241400,1183780040663220 %N A285044 Expansion of cosh(5*arctanh(2*sqrt(x))). %C A285044 Note that the function cosh(2*n*arctanh(sqrt(x))) is the o.g.f. for the coordination sequence of the C_n lattice. See, for example, A010006. %C A285044 In A285043 through A285046 we consider sequences with o.g.f. cosh((2*n+1)*arctanh(2*sqrt(x))) for n = 1, 2, 3 and 4: n = 0 gives the central binomial coefficients A000984. %F A285044 a(n) = 1/3*(64*n^2 + 8*n + 3)*binomial(2*n,n). %F A285044 O.g.f. cosh(5*arctanh(2*sqrt(x))) = (1 + 40*x + 80*x^2)/(1 - 4*x)^(5/2) = 1 + 50*x + 550*x^2 + 4020*x^3 + .... %F A285044 Note that the zeros of the polynomial 1 + 40*x^2 + 80*x^4 = 1/2*((1 + 2*x)^5 + (1 - 2*x)^5), are given by 1/2*cot(k*Pi/5)*i for 1 <= k <= 4. See A085840. %F A285044 O.g.f. for the sequence with interpolated zeros: 1/2*( ((1 + 2*x)/(1 - 2*x))^(5/2) + ((1 - 2*x)/(1 + 2*x))^(5/2) ) = 1 + 50*x^2 + 550*x^4 + 4020*x^6 + .... %p A285044 seq(1/3*(64*n^2 + 8*n + 3)*binomial(2*n,n), n = 0..20); %Y A285044 Cf. A000984, A010006, A085840, A285043, A285045, A285046. %K A285044 nonn,easy %O A285044 0,2 %A A285044 _Peter Bala_, Apr 10 2017