A144703 -id:A144703 - cp's OEIS frontend

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

Alois P. Heinz, Sep 19 2008

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.

			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
...

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.

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

Denominators of S(n,k): A144703.

Cf. A144815, A144816.

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);

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 *)

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

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.

cp's OEIS Frontend

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.

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