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.

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

Author

Alois P. Heinz, Sep 21 2008

Keywords

• frac
• sign
• tabl
• look

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

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

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.