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A244421 Denominators of coefficient triangle for expansion of x^n in terms of polynomials Todd(k,x) = T(2*k+1, sqrt(x))/sqrt(x) (A084930), with the Chebyshev T-polynomials.

%I A244421 #12 Mar 16 2025 02:56:17
%S A244421 1,4,4,8,16,16,64,64,64,64,128,64,64,256,256,512,512,1024,1024,1024,
%T A244421 1024,1024,4096,4096,2048,2048,4096,4096,16384,16384,16384,16384,
%U A244421 16384,16384,16384,16384,32768,8192,8192,16384,16384,8192,8192,65536,65536,131072,131072,65536,65536,65536,65536,262144,262144,262144,262144,262144,524288,524288,131072,131072,1048576,1048576,524288,524288,1048576,1048576
%N A244421 Denominators of coefficient triangle for expansion of x^n in terms of polynomials Todd(k,x) = T(2*k+1, sqrt(x))/sqrt(x) (A084930), with the Chebyshev T-polynomials.
%C A244421 For the numerator triangle see A244420, also for comments, and the rational entries R(n,m) of the lower triangular Riordan matrix denoted in standard fashion by ((2 - c(z/4))/(1-z), -1 + c(z/4)) with c the o.g.f. of the Catalan numbers A000108.
%H A244421 Wolfdieter Lang, <a href="/A244421/a244421.pdf">First rows of the triangle.</a>
%F A244421 a(n,m) = denominator(R(n,m)) with the rationals Riordan matrix elements R(n,m)= [x^m]R(n,x), with the row polynomials R(n,x) generated by ((2 - c(z/4))/(1-z))/(1 - x*(-1 + c(z/4))) = 2*((1+x)*(z-1) + (1-x)*sqrt(1-z))/((1-z)*((1+x)^2*z - 4*x)), where c(x) is the o.g.f. of the Catalan numbers A000108.
%e A244421 The triangle a(n,m) begins:
%e A244421   n\m   0     1     2     3     4     5     6 ...
%e A244421   0:    1
%e A244421   1:    4     4
%e A244421   2:    8    16    16
%e A244421   3:   64    64    64    64
%e A244421   4:  128    64    64   256   256
%e A244421   5:  512   512  1024  1024  1024  1024
%e A244421   6: 1024  4096  4096  2048  2048  4096  4096
%e A244421   ...
%e A244421 For more rows see the link.
%e A244421 For the rational triangle R(n,m) see the example section of A244420.
%e A244421 Expansion: x^3 = (35*Todd(0, x) + 21*Todd(1, x) + 7*Todd(2, x) + 1*Todd(3, x))/64 = (35 + 21*(-3+4*x) + 7*( 5-20*x+16*x^2) + (-7+56*x-112*x^2+64*x^3))/64. For the Todd polynomials see A084930.
%Y A244421 Cf. A084930, A244420, A000108.
%K A244421 nonn,easy,frac,tabl
%O A244421 0,2
%A A244421 _Wolfdieter Lang_, Aug 04 2014