A230142 Numerator of 1/u_n(1/2), where polynomial u_n(x) is used to approximate x->sin(Pi*x)/Pi.

16, 256, 5632, 1048576, 80543744, 18185977856, 2823575044096, 4608812904349696, 1194823452775677952, 766890677854432919552, 298370458295691856838656, 184465173199612912007643136, 301475731054794304317414178816, 381273851270136749855228154609664

Offset: 1

Author

Alois P. Heinz, Oct 10 2013

Keywords

• nonn
• frac

Comments

Coefficients of u_n are given by the n-th row of A144846/A144847.

Examples

			16/3, 256/75, 5632/1785, 1048576/333795, 80543744/25638459, 18185977856/5788790007, 2823575044096/898772045457 ... = A230142/A230143
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..99

Crossrefs

Cf. A000796.

Programs

Maple

u:= proc(n) option remember; local f, i, x; f:= unapply(simplify(sum('cat(a||(2*i+1)) *x^(2*i+1)', 'i'=0..n) ), x); unapply(subs(solve({f(1)=0, seq((D@@i)(f)(1)=`if`(i=1, -1, -(D@@i)(f)(0)), i=1..n)}, {seq(cat(a||(2*i+1)), i=0..n)}), sum('cat(a||(2*i+1)) *x^(2*i+1)', 'i'=0..n)), x) end: seq(numer(1/u(n)(1/2)), n=1..15);

Mathematica

u[n_] := u[n] = Module[{f, i, x, a}, f = Function[x, Sum[a[2*i+1]*x^(2*i+1), {i, 0, n}]]; Function[x, Sum[a[2*i+1]*x^(2*i+1), {i, 0, n}] /. First @ Solve[Join[{f[1] == 0}, Table[Derivative[i][f][1] == If[i == 1, -1, -Derivative[i][f][0]], {i, 1, n}]], Table[a[2*i+1], {i, 0, n}]]]]; Table[Numerator[1/u[n][1/2]], {n, 1, 15}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

limit_{n->infinity} 1/u_n(1/2) = Pi.