cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A144860 Denominators of triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the coefficient of x^(2k+1) in polynomial v_n(x), used to approximate x->sin(Pi*x)/Pi.

Original entry on oeis.org

1, 1, 1, 1, 7, 7, 1, 87, 29, 87, 1, 2047, 2047, 2047, 2047, 1, 15731, 78655, 15731, 15731, 78655, 1, 4439935, 887987, 4439935, 887987, 4439935, 4439935, 1, 49239241, 49239241, 344674687, 49239241, 49239241, 49239241, 344674687, 1, 35162451967
Offset: 0

Views

Author

Alois P. Heinz, Sep 23 2008

Keywords

Links

Crossrefs

See A144859 for more information on T(n,k). Column k=0 gives: A000012.

Programs

  • Maple
    seq(seq(denom(T(n,k)), k=0..n), n=0..9);

A230144 Numerator of 1/v_n(1/2), where polynomial v_n(x) is used to approximate x->sin(Pi*x)/Pi.

Original entry on oeis.org

8, 224, 1856, 1048064, 80542720, 18185973760, 2823575035904, 4608812904218624, 398274484258471936, 766890677854431870976, 298370458295691854741504, 553395519598838736006152192, 301475731054794304317380624384, 381273851270136749855228020391936
Offset: 1

Views

Author

Alois P. Heinz, Oct 10 2013

Keywords

Comments

Coefficients of v_n are given by the n-th row of A144859/A144860.

Examples

			8/3, 224/75, 1856/595, 1048064/333795, 80542720/25638459, 18185973760/5788790007, 2823575035904/898772045457 ... = A230144/A230145

Links

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.

A230145 Denominator of 1/v_n(1/2), where polynomial v_n(x) is used to approximate x->sin(Pi*x)/Pi.

Original entry on oeis.org

3, 75, 595, 333795, 25638459, 5788790007, 898772045457, 1467030741832227, 126774706022852173, 244108884436744360605, 94974266622893811200463, 176151264858556860995936775, 95962705639251788100721754775, 121363236202656183485569513082175
Offset: 1

Views

Author

Alois P. Heinz, Oct 10 2013

Keywords

Comments

Coefficients of v_n are given by the n-th row of A144859/A144860.

Links

Crossrefs

Numerators are given in A230144.
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(denom(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[Denominator[1/v[n][1/2]], {n, 1, 15}] (* Jean-François Alcover, Feb 13 2014, after Maple *)
Showing 1-3 of 3 results.

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