OFFSET
1,1
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
KEYWORD
nonn,frac
AUTHOR
Alois P. Heinz, Oct 10 2013
STATUS
approved