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%I #33 Jul 20 2023 16:28:57
%S 1,3,-1,15,-5,3,35,-35,21,-5,315,-105,189,-45,35,693,-1155,693,-495,
%T 385,-63,3003,-3003,9009,-2145,5005,-819,231,6435,-15015,27027,-32175,
%U 25025,-12285,3465,-429,109395,-36465,153153,-109395,425425,-69615,58905,-7293,6435
%N 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.
%C 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.
%C 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.
%H Alois P. Heinz, <a href="/A144815/b144815.txt">Rows n = 0..140, flattened</a>
%H Alois P. Heinz, <a href="/A144815/a144815.gif">Animation of sigma_n(x) and their derivatives for n=0..15</a>
%F See program.
%e 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
%e As triangle:
%e 1;
%e 3/2, -1/2;
%e 15/8, -5/4, 3/8;
%e 35/16, -35/16, 21/16, -5/16;
%e 315/128, -105/32, 189/64, -45/32, 35/128;
%e ...
%p 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);
%t 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 *)
%t 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 *)
%Y Denominators of T(n,k): A144816.
%Y Column k=0 gives A001803.
%Y Diagonal gives (-1)^n A001790(n).
%Y Cf. A144702, A144703.
%K frac,sign,tabl,look
%O 0,2
%A _Alois P. Heinz_, Sep 21 2008
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