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A244421
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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.
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5
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1, 4, 4, 8, 16, 16, 64, 64, 64, 64, 128, 64, 64, 256, 256, 512, 512, 1024, 1024, 1024, 1024, 1024, 4096, 4096, 2048, 2048, 4096, 4096, 16384, 16384, 16384, 16384, 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
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OFFSET
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0,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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The triangle a(n,m) begins:
n\m 0 1 2 3 4 5 6 ...
0: 1
1: 4 4
2: 8 16 16
3: 64 64 64 64
4: 128 64 64 256 256
5: 512 512 1024 1024 1024 1024
6: 1024 4096 4096 2048 2048 4096 4096
...
For more rows see the link.
For the rational triangle R(n,m) see the example section of A244420.
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.
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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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