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 A085840 #19 Oct 19 2017 03:14:20
%S A085840 1,1,12,1,40,80,1,84,560,448,1,144,2016,5376,2304,1,220,5280,29568,
%T A085840 42240,11264,1,312,11440,109824,329472,292864,53248,1,420,21840,
%U A085840 320320,1647360,3075072,1863680,245760
%N A085840 Triangle read by rows: T(n,m) = 4^m * (2*n+1)! / ( (2*n - 2*m + 1)! * (2*m)! ), row n has n+1 terms.
%C A085840 Row n has the unsigned coefficients of a polynomial whose roots are 2*tan(Pi*k/(2n+1)) [for k = 1 to 2n].
%C A085840 Polynomial of row n = Sum_{m=0..n} (-1)^m T(n,m) x^(2n-2m).
%F A085840 From _Peter Bala_, Apr 10 2017: (Start)
%F A085840 O.g.f.: (1 - (1 - 4*x)*t)/(1 - 2*(1 + 4*x)*t + (1 - 4*x)^2*t^2) = 1 + (1 + 12*x)*t + (1 + 40*x + 80*x^2)*t^2 + ....
%F A085840 n_th row polynomial R(n,x) = 1/2*( (1 + 2*sqrt(x))^(2*n+1) + (1 - 2*sqrt(x))^(2*n+1) ). These polynomials occur in the expansion cosh((2*n + 1)*arctanh(2*x)) = R(n,x^2)/(1 - 4*x^2)^(n+1/2). See A285043 - A285046.
%F A085840 For n >= 1, R(n,x) = (1 - 4*x)^n( U(n,(1 + 4*x)/(1 - 4*x)) - U(n-1,(1 + 4*x)/(1 - 4*x)) ), where U(n,x) is the n-th Chebyshev polynomial of the second kind. (End)
%e A085840 1
%e A085840 x^2 - 12
%e A085840 x^4 - 40x^2 + 80
%e A085840 x^6 - 84x^4 + 560x^2 - 448
%e A085840 x^8 - 144x^6 + 2016x^4 - 5376x^2 + 2304
%e A085840 x^10 - 220x^8 + 5280x^6 - 29568x^4 + 42240x^2 - 11264
%e A085840 Polynomial #4 has eight roots: 2 tan (Pi*k/9) for k=1 to 8.
%p A085840 for n from 0 to 10 do lprint(seq(4^k*binomial(2*n + 1, 2*k), k = 0..n)) end do; # _Peter Bala_, Apr 10 2017
%Y A085840 Cf. A085841, A285043, A284044, A285045, A285046.
%K A085840 nonn,tabl,easy
%O A085840 0,3
%A A085840 _Gary W. Adamson_, Jul 05 2003
%E A085840 Edited by _Don Reble_, Nov 13 2005