A144816
Denominators of triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the coefficient of x^(2*k+1) in polynomial t_n(x), used to define continuous and n times differentiable sigmoidal transfer functions.
Original entry on oeis.org
1, 2, 2, 8, 4, 8, 16, 16, 16, 16, 128, 32, 64, 32, 128, 256, 256, 128, 128, 256, 256, 1024, 512, 1024, 256, 1024, 512, 1024, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 2048, 32768, 4096, 8192, 4096, 16384, 4096, 8192, 4096, 32768, 65536, 65536, 16384, 16384, 32768, 32768, 16384, 16384, 65536, 65536
Offset: 0
Triangle begins:
1;
2, 2;
8, 4, 8;
16, 16, 16, 16;
128, 32, 64, 32, 128;
...
See
A144815 for more information on T(n,k).
Main diagonal and column k=0 gives
A046161.
-
# Function T(n,k) defined in A144815.
seq(seq(denom(T(n,k)), k=0..n), n=0..10);
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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] // Denominator, {n, 0, 10}] // Flatten (* Jean-François Alcover, Feb 03 2014 *)
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.
Original entry on oeis.org
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
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.
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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);
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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 *)
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