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.

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

Author

Alois P. Heinz, Sep 19 2008

Keywords

• frac
• tabf
• sign
• look

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

• Alois P. Heinz, Rows n = 0..114, flattened
• Alois P. Heinz, Animation of sigma_n(x) and their derivatives for n=0..15

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.