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A244420
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Numerators 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 polynomials of the first kind (type T).
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5
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1, 3, 1, 5, 5, 1, 35, 21, 7, 1, 63, 21, 9, 9, 1, 231, 165, 165, 55, 11, 1, 429, 1287, 715, 143, 39, 13, 1, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 12155, 2431, 1547, 1547, 595, 85, 17, 17, 1, 46189, 37791, 12597, 6783, 2907, 969, 969, 171, 19, 1, 88179, 146965, 101745, 14535, 6783, 20349, 5985, 665, 105, 21, 1
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OFFSET
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0,2
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COMMENTS
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This expansion is due to the Riordan property of the triangle A084930. The inverse of the lower triangular matrix built by A084930 is therefore also a (rational) Riordan triangle, namely ((2 - c(z/4)/(1-z), -1 + c(z/4)) in the standard notation, where c is the o.g.f. of A000108 (Catalan).
For the denominators of this triangle see A244421.
The expansion is x^n = sum(R(n,m)*Todd(m, x), m=0..n), n >= 0, with the rational triangle with entries R(n,m) = a(n, m)/b(n, m) with b(n, m) = A244421(n, m).
If one uses instead the expansion of (4*x)^n one finds the integer triangle A111418: (4*x)^n = sum(A111418(n,k) * Todd(k, x), k=0..n).
The row sums of the rational triangle R(n,m) are identically 1. The alternating row sums have o.g.f. 1/sqrt(1-x) which generates A001790(n)/A046161(n) (see a Michael Somos comment on A046161), namely 1, 1/2, 3/8, 5/16, 35/128, 63/256, 231/1024, 429/2048, ...
R(n,m) = a(n, m)/ A244421(n, m) is also the rational triangle for the expansion cos^{2*n+1}(x) = Sum_{m=0..n} R(n, m)*cos((2*m+1)*x), n >= 0, m = 0..n. Compare with the odd numbered rows of A273496. In terms of Chebyshev T-polynomials (A053120) this is the identity x^(2*n+1) = Sum_{m=0..n} R(n, m)*T(2*m+1, x).
S(n,m) = (-1)^m*a(n, m)/ A244421(n, m) is the rational triangle for the expansion sin^{2*n+1}(x) = Sum_{m=0..n} S(n, m)*sin((2*m+1)*x), n >= 0, m = 0..n. In terms of Chebyshev S-polynomials (A049310) this is equivalent to the identity (4 - x^2)*n = Sum_{m=0..n} (-1)^m * binomial(n, n-m)*S(2*m,x), n >= 0.
(End)
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LINKS
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FORMULA
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a(n, m) = numerator(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.
The rationals R(n, m) = binomial(2*n+1, m)/2^(2*n). - Wolfdieter Lang, Jun 12 2016
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EXAMPLE
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The numerator triangle a(n,m) begins:
n\m 0 1 2 3 4 5 6 7 8 9
0: 1
1: 3 1
2: 5 5 1
3: 35 21 7 1
4: 63 21 9 9 1
5: 231 165 165 55 11 1
6: 429 1287 715 143 39 13 1
7: 6435 5005 3003 1365 455 105 15 1
8: 12155 2431 1547 1547 595 85 17 17 1
9: 46189 37791 12597 6783 2907 969 969 171 19 1
---
The rational triangle R(n,m) begins:
n\m 0 1 2 3 4 5
0: 1
1: 3/4 1/4
2: 5/8 5/16 1/16
3: 35/64 21/64 7/64 1/64
4: 63/128 21/64 9/64 9/256 1/256
5: 231/512 165/512 165/1024 55/1024 11/1024 1/1024
...
The next rows are:
n=8: 12155/32768, 2431/8192, 1547/8192, 1547/16384, 595/16384, 85/8192, 17/8192,
17/65536, 1/65536,
n=9: 46189/131072, 37791/131072, 12597/65536, 6783/65536, 2907/65536, 969/65536, 969/262144, 171/262144, 19/262144, 1/262144,
n=10: 88179/262144, 146965/524288, 101745/524288, 14535/131072, 6783/131072, 20349/1048576, 5985/1048576, 665/524288, 105/524288, 21/1048576, 1/1048576.
...
Expansions:
x^2 = 5/8 * Todd(0,x) + 5/16 * Todd(1,x) + 1/16 * Todd(2,x) = 5/8 + (5/16)*(-3 + 4*x) +(1/16)*(5 -20*x + 16*x^2).
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 the coefficient table 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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