A144702 -id:A144702 - cp's OEIS frontend

A144703 Denominators 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.

2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 4, 1, 2, 2, 4, 2, 16, 1, 16, 1, 16, 2, 16, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 16, 1, 4, 1, 8, 1, 4, 2, 16, 1, 2, 32, 1, 32, 1, 16, 1, 16, 2, 32, 1, 32, 2, 256, 1, 128, 1, 256, 1, 64, 1, 256, 1, 128, 2, 256, 2, 128, 1, 64, 1, 128, 1, 32, 1, 128, 2, 64, 2, 128, 2, 2

Offset: 0

Alois P. Heinz, Sep 19 2008

Alois P. Heinz, Rows n = 0..114, flattened

See A144702 for more information on S(n,k).

seq(seq(denom(coeff(s(n)(x), x, k)), k=0..ceil((3*n+1)/2)), n=0..10);

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

Alois P. Heinz, Sep 21 2008

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.

			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;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Alois P. Heinz, Animation of sigma_n(x) and their derivatives for n=0..15

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

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);

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 *)

See program.

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A144703 Denominators 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.

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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.

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