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 1-4 of 4 results.

A144847 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 u_n(x), used to approximate x->sin(Pi*x)/Pi.

Original entry on oeis.org

1, 2, 2, 8, 4, 8, 88, 22, 88, 44, 2048, 64, 1024, 256, 2048, 78656, 4916, 19664, 39328, 78656, 39328, 4439936, 34687, 277496, 1109984, 4439936, 1109984, 2219968, 344674688, 2692771, 21542168, 43084336, 344674688, 86168672, 172337344
Offset: 0

Views

Author

Alois P. Heinz, Sep 22 2008

Keywords

Links

Crossrefs

See A144846 for more information on T(n,k).

Programs

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

A144859 Numerators 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

0, 1, -1, 1, -10, 3, 1, -140, 21, -10, 1, -3360, 1638, -360, 35, 1, -25872, 63756, -2970, 385, -126, 1, -7303296, 720720, -845988, 23023, -9828, 462, 1, -80995200, 39969072, -65739960, 1286285, -114660, 6930, -1716, 1, -57839907840
Offset: 0

Views

Author

Alois P. Heinz, Sep 23 2008

Keywords

Comments

All even coefficients of v_n are 0. Sum_{k=0..n} T(n,k) = 0. 1/v(n)(1/2) is an approximation to Pi, cf. A230144/A230145. D(v_n)(0) = 1 if n>0.

Examples

			0, 1, -1, 1, -10/7, 3/7, 1, -140/87, 21/29, -10/87, 1, -3360/2047, 1638/2047, -360/2047, 35/2047, 1, -25872/15731, 63756/78655, -2970/15731, 385/15731, -126/78655 ... = A144859/A144860
As triangle:
  0
  1,   -1
  1,  -10/7,   3/7
  1, -140/87, 21/29, -10/87

Links

Crossrefs

Denominators of T(n,k): A144860. Diagonal gives: A110556(n) for n>0 and (-1)^n A001700(n-1) for n>0. First column gives: A057427. Cf. A144846.

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: T:= (n,k)-> coeff(v(n)(x), x, 2*k+1): seq(seq(numer(T(n,k)), k=0..n), n=0..9);
  • Mathematica
    v[n_] := v[n] = Module[{f, i, x, a}, f[x_] = Sum[a[2*i+1]*x^(2i+1), {i, 0, n}]; Function[x, Sum[a[2*i+1]*x^(2i+1), {i, 0, n}] /. First @ Solve [{f[1] == 0, If[n == 0, True, f'[0] == 1], Sequence @@ Table[Derivative[i][f][1] == -Derivative[i][f][0], {i, 2, n}]}, Table[a[2*i+1], {i, 0, n}]]]]; T[n_, k_] := Coefficient[v[n][x], x, 2*k+1]; Table[Table[Numerator[T[n, k]], {k, 0, n}], {n, 0, 9}] // Flatten (* Jean-François Alcover, Feb 12 2014, translated from Maple *)

Formula

See program.

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

Original entry on oeis.org

16, 256, 5632, 1048576, 80543744, 18185977856, 2823575044096, 4608812904349696, 1194823452775677952, 766890677854432919552, 298370458295691856838656, 184465173199612912007643136, 301475731054794304317414178816, 381273851270136749855228154609664
Offset: 1

Views

Author

Alois P. Heinz, Oct 10 2013

Keywords

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

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.

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

Original entry on oeis.org

3, 75, 1785, 333795, 25638459, 5788790007, 898772045457, 1467030741832227, 380324118068556519, 244108884436744360605, 94974266622893811200463, 58717088286185620331978925, 95962705639251788100721754775, 121363236202656183485569513082175
Offset: 1

Views

Author

Alois P. Heinz, Oct 10 2013

Keywords

Comments

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

Links

Crossrefs

Numerators are given in A230142.
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(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 *)

Formula

Limit_{n->infinity} 1/u_n(1/2) = Pi.
Showing 1-4 of 4 results.

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