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A144702
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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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3
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,12
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COMMENTS
| A sigmiodal 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.
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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.
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LINKS
| Alois P. Heinz, Rows n = 0..114, flattened
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FORMULA
| See program.
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EXAMPLE
| 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
...
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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);
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CROSSREFS
| Denominators of S(n, k): A144703.
Sequence in context: A199382 A140240 A091672 * A156716 A055510 A200397
Adjacent sequences: A144699 A144700 A144701 * A144703 A144704 A144705
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KEYWORD
| frac,tabf,sign
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AUTHOR
| Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 19 2008
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