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.

A077070 Triangle read by rows: T(n,k) is the power of 2 in denominator of coefficients of Legendre polynomials, where n >= 0 and 0 <= k <= n.

Original entry on oeis.org

0, 1, 1, 3, 2, 3, 4, 4, 4, 4, 7, 5, 6, 5, 7, 8, 8, 7, 7, 8, 8, 10, 9, 10, 8, 10, 9, 10, 11, 11, 11, 11, 11, 11, 11, 11, 15, 12, 13, 12, 14, 12, 13, 12, 15, 16, 16, 14, 14, 15, 15, 14, 14, 16, 16, 18, 17, 18, 15, 17, 16, 17, 15, 18, 17, 18, 19, 19, 19, 19, 18, 18, 18, 18, 19, 19, 19, 19, 22, 20, 21, 20, 22, 19, 20, 19, 22, 20, 21, 20, 22
Offset: 0

Views

Author

Michael Somos, Oct 25 2002

Keywords

Examples

			Triangle T(n,k) (with rows n >= 0 and columns k >= 0) begins as follows:
   0;
   1,  1;
   3,  2,  3;
   4,  4,  4,  4;
   7,  5,  6,  5,  7;
   8,  8,  7,  7,  8,  8;
  10,  9, 10,  8, 10,  9, 10;
  ...

Links

Crossrefs

Cf. A005187 (column k=0), A101925 (column k=1), A077071 (row sums), A144816 (denominators).
Cf. A000120, A007814.

Programs

  • Maple
    T:= n-> (p-> seq(padic[ordp](denom(coeff(p, x, i)), 2)
             , i=0..2*n, 2))(orthopoly[P](2*n, x)):
seq(T(n), n=0..12);  # Alois P. Heinz, Jan 25 2022
  • Mathematica
    T[n_, k_] := IntegerExponent[Denominator[Coefficient[LegendreP[2n, x], x, 2k]], 2]; Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2017 *)
  • PARI
    {T(n, k) = if( k<0 || k>n, 0, -valuation( polcoeff( pollegendre(2*n), 2*k), 2))}
  • PARI
    T(n,k) = 2*n - hammingweight(n-k) - hammingweight(k); \\ Kevin Ryde, Jan 29 2022

Formula

T(n, k) = A007814(A144816(n, k)). - Michel Marcus, Jan 29 2022
T(n, k) = 2*n - wt(n-k) - wt(k) where wt = A000120 is the binary weight. - Kevin Ryde, Jan 29 2022

A144702 Numerators of triangle S(n,k), n>=0, 0<=k<=ceiling((3n+1)/2): S(n,k) is the coefficient of x^k in polynomial s_n(x), used to define continuous and n times differentiable sigmoidal transfer functions.

Original entry on oeis.org

1, 1, 1, 1, -1, 1, 1, 0, -1, 1, 1, 5, 0, -5, 5, -3, 1, 21, 0, -35, 0, 63, -7, 15, 1, 3, 0, -7, 0, 21, -14, 15, -3, 1, 25, 0, -15, 0, 63, 0, -75, 45, -175, 2, 1, 55, 0, -165, 0, 231, 0, -825, 165, -1925, 22, -105, 1, 455, 0, -715, 0, 3861, 0, -2145, 0, 25025, -143, 12285, -65
Offset: 0

Views

Author

Alois P. Heinz, Sep 19 2008

Keywords

Comments

A sigmoidal transfer function sigma_n: R->[0,1] can be defined as sigma_n(x) = 1 if x>1, sigma_n(x) = s_n(x) if x in [0,1] and sigma_n(x) = 1-sigma_n(-x) if x<0.

Examples

			1/2, 1/2, 1/2, 1, -1/2, 1/2, 1, 0, -1, 1/2, 1/2, 5/4, 0, -5/2, 5/2, -3/4, 1/2, 21/16, 0, -35/16, 0, 63/16, -7/2, 15/16, 1/2, 3/2, 0, -7/2, 0, 21/2, -14, 15/2, -3/2 ... = A144702/A144703
As triangle:
1/2   1/2
1/2   1     -1/2
1/2   1      0     -1     1/2
1/2   5/4    0     -5/2   5/2  -3/4
1/2  21/16   0    -35/16  0    63/16   -7/2   15/16
1/2   3/2    0     -7/2   0    21/2   -14     15/2   -3/2
1/2  25/16   0    -15/4   0    63/8     0    -75/4   45/2  -175/16  2
...

References

  • A. P. Heinz: Yes, trees may have neurons. In Computer Science in Perspective, R. Klein, H. Six and L. Wegner, Editors Lecture Notes In Computer Science 2598. Springer-Verlag New York, New York, NY, 2003, pages 179-190.

Links

Crossrefs

Denominators of S(n,k): A144703.
Cf. A144815, A144816.

Programs

  • Maple
    s:= proc(n) option remember; local t,u,f,i,x; u:= floor(n/2); t:= u+n+1; f:= unapply(simplify(1/2 +sum('cat(a||i) *x^i', 'i'=1..t) -sum('cat(a||(2*i)) *x^(2*i)', 'i'=1..u)), x); unapply(subs(solve({f(1)=1, seq((D@@i)(f)(1)=0, i=1..n)}, {seq(cat(a||i), i=1..t)}), 1/2 +sum('cat(a||i) *x^i', 'i'=1..t) -sum('cat(a||(2*i)) *x^(2*i)', 'i'=1..u)), x); end: seq(seq(numer(coeff(s(n)(x), x,k)), k=0..ceil((3*n+1)/2)), n=0..10);
  • Mathematica
    s[n_] := s[n] = Module[{t, u, f, i, x, a}, u = Floor[n/2]; t = u+n+1; f = Function[x, 1/2+Sum[a[i]*x^i, {i, 1, t}] - Sum[a[2*i]*x^(2i), {i, 1, u}]]; Function[x, 1/2+Sum[a[i]*x^i, {i, 1, t}] - Sum[a[2*i]*x^(2i), {i, 1, u}] /. First @ Solve[{f[1] == 1, Sequence @@ Table[Derivative[i][f][1] == 0, {i, 1, n}]}]]]; Table[Table[Numerator[Coefficient[s[n][x], x, k]], {k, 0, Ceiling[(3*n+1)/2]}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 13 2014, after Maple *)

Formula

See program.

A144815 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 t_n(x), used to define continuous and n times differentiable sigmoidal transfer functions.

Original entry on oeis.org

1, 3, -1, 15, -5, 3, 35, -35, 21, -5, 315, -105, 189, -45, 35, 693, -1155, 693, -495, 385, -63, 3003, -3003, 9009, -2145, 5005, -819, 231, 6435, -15015, 27027, -32175, 25025, -12285, 3465, -429, 109395, -36465, 153153, -109395, 425425, -69615, 58905, -7293, 6435
Offset: 0

Views

Author

Alois P. Heinz, Sep 21 2008

Keywords

Comments

All even coefficients of t_n have to be 0, because t_n is defined to be point-symmetric with respect to the origin, with vanishing n-th derivative for x=1.
A sigmoidal transfer function sigma_n: R->[ -1,1] can be defined as sigma_n(x) = 1 if x>1, sigma_n(x) = t_n(x) if x in [ -1,1] and sigma_n(x) = -1 if x<-1.

Examples

			1, 3/2, -1/2, 15/8, -5/4, 3/8, 35/16, -35/16, 21/16, -5/16, 315/128, -105/32, 189/64, -45/32, 35/128, 693/256, -1155/256, 693/128, -495/128, 385/256, -63/256 ... = A144815/A144816
As triangle:
    1;
    3/2,     -1/2;
   15/8,     -5/4,    3/8;
   35/16,   -35/16,  21/16,  -5/16;
  315/128, -105/32, 189/64, -45/32, 35/128;
  ...

Links

Crossrefs

Denominators of T(n,k): A144816.
Column k=0 gives A001803.
Diagonal gives (-1)^n A001790(n).
Cf. A144702, A144703.

Programs

  • Maple
    t:= 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)=1, seq((D@@i)(f)(1)=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: T:= (n,k)-> coeff(t(n)(x), x, 2*k+1): seq(seq(numer(T(n,k)), k=0..n), n=0..10);
  • Mathematica
    row[n_] := Module[{f, a, eq}, f = Function[x, Sum[a[2*k+1]*x^(2*k+1), {k, 0, n}]]; eq = Table[Derivative[k][f][1] == If[k == 0, 1, 0], {k, 0, n}]; Table[a[2*k+1], {k, 0, n}] /. Solve[eq] // First]; Table[row[n] // Numerator, {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 03 2014 *)
Flatten[Table[Numerator[CoefficientList[Hypergeometric2F1[1/2,1-n,3/2,x^2]*(2*n)!/(n!*(n-1)!*2^(2*n-1)),x^2]],{n,1,9}]] (* Eugeniy Sokol, Aug 20 2019 *)

Formula

See program.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.