A230143 Denominator of 1/u_n(1/2), where polynomial u_n(x) is used to approximate x->sin(Pi*x)/Pi.

3, 75, 1785, 333795, 25638459, 5788790007, 898772045457, 1467030741832227, 380324118068556519, 244108884436744360605, 94974266622893811200463, 58717088286185620331978925, 95962705639251788100721754775, 121363236202656183485569513082175

Offset

1,1

Comments

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

Links

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

Formula

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

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(denom(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[Denominator[1/u[n][1/2]], {n, 1, 15}] (* Jean-François Alcover, Feb 13 2014, after Maple *)

Crossrefs

Numerators are given in A230142.

Cf. A000796.

Sequence in context: A012491 A136328 A003690 * A228841 A361537 A249938

Adjacent sequences: A230140 A230141 A230142 * A230144 A230145 A230146

Keyword

nonn,frac

Author

Alois P. Heinz, Oct 10 2013

Status

approved