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A144815
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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.
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3
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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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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See program.
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EXAMPLE
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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;
...
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MAPLE
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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);
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MATHEMATICA
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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] // 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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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