A230144 Numerator of 1/v_n(1/2), where polynomial v_n(x) is used to approximate x->sin(Pi*x)/Pi.
8, 224, 1856, 1048064, 80542720, 18185973760, 2823575035904, 4608812904218624, 398274484258471936, 766890677854431870976, 298370458295691854741504, 553395519598838736006152192, 301475731054794304317380624384, 381273851270136749855228020391936
Offset: 1
Examples
8/3, 224/75, 1856/595, 1048064/333795, 80542720/25638459, 18185973760/5788790007, 2823575035904/898772045457 ... = A230144/A230145
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..99
Crossrefs
Cf. A000796.
Programs
-
Maple
v:= 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, `if`(n=0, NULL, D(f)(0)=1), seq((D@@i)(f)(1)=-(D@@i)(f)(0), i=2..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/v(n)(1/2)), n=1..15); -
Mathematica
v[n_] := v[n] = Module[{f, i, x, a}, f[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}, {If[n == 0, True, f'[0] == 1]}, Table[Derivative[i][f][1] == -Derivative[i][f][0], {i, 2, n}]]]]]; Table[Numerator[1/v[n][1/2]], {n, 1, 15}] (* Jean-François Alcover, Feb 13 2014, after Maple *)
Formula
limit_{n->infinity} 1/v_n(1/2) = Pi.
Comments