A144702 - OEIS

%I #34 Jul 19 2023 18:56:49

%S 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,

%T 21,-14,15,-3,1,25,0,-15,0,63,0,-75,45,-175,2,1,55,0,-165,0,231,0,

%U -825,165,-1925,22,-105,1,455,0,-715,0,3861,0,-2145,0,25025,-143,12285,-65

%N 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.

%C 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.

%D 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.

%H Alois P. Heinz, <a href="/A144702/b144702.txt">Rows n = 0..114, flattened</a>

%H Alois P. Heinz, <a href="/A144702/a144702.gif">Animation of sigma_n(x) and their derivatives for n=0..15</a>

%F See program.

%e 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

%e As triangle:

%e 1/2   1/2

%e 1/2   1     -1/2

%e 1/2   1      0     -1     1/2

%e 1/2   5/4    0     -5/2   5/2  -3/4

%e 1/2  21/16   0    -35/16  0    63/16   -7/2   15/16

%e 1/2   3/2    0     -7/2   0    21/2   -14     15/2   -3/2

%e 1/2  25/16   0    -15/4   0    63/8     0    -75/4   45/2  -175/16  2

%e ...

%p 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);

%t 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 *)

%Y Denominators of S(n,k): A144703.

%Y Cf. A144815, A144816.

%K frac,tabf,sign,look

%O 0,12

%A _Alois P. Heinz_, Sep 19 2008