OFFSET
0,2
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.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
Alois P. Heinz, Animation of sigma_n(x) and their derivatives for n=0..15
FORMULA
See program.
EXAMPLE
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 *)
CROSSREFS
AUTHOR
Alois P. Heinz, Sep 21 2008
STATUS
approved