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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.

%I A144816 #30 Oct 08 2024 20:06:32
%S A144816 1,2,2,8,4,8,16,16,16,16,128,32,64,32,128,256,256,128,128,256,256,
%T A144816 1024,512,1024,256,1024,512,1024,2048,2048,2048,2048,2048,2048,2048,
%U A144816 2048,32768,4096,8192,4096,16384,4096,8192,4096,32768,65536,65536,16384,16384,32768,32768,16384,16384,65536,65536
%N 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.
%H A144816 Alois P. Heinz, <a href="/A144816/b144816.txt">Rows n = 0..140, flattened</a>
%e A144816 Triangle begins:
%e A144816     1;
%e A144816     2,  2;
%e A144816     8,  4,  8;
%e A144816    16, 16, 16, 16;
%e A144816   128, 32, 64, 32, 128;
%e A144816   ...
%p A144816 # Function T(n,k) defined in A144815.
%p A144816 seq(seq(denom(T(n,k)), k=0..n), n=0..10);
%t A144816 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 *)
%Y A144816 See A144815 for more information on T(n,k).
%Y A144816 Main diagonal and column k=0 gives A046161.
%Y A144816 Column k=1 gives A101926(n-1) = 2^A101925(n-1) = 2^(A005187(n-1)+1).
%Y A144816 Cf. A077070.
%K A144816 frac,nonn,tabl,look
%O A144816 0,2
%A A144816 _Alois P. Heinz_, Sep 21 2008