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A244420 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).

%I A244420 #29 Mar 16 2025 02:34:45
%S A244420 1,3,1,5,5,1,35,21,7,1,63,21,9,9,1,231,165,165,55,11,1,429,1287,715,
%T A244420 143,39,13,1,6435,5005,3003,1365,455,105,15,1,12155,2431,1547,1547,
%U A244420 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
%N A244420 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).
%C A244420 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).
%C A244420 For the denominators of this triangle see A244421.
%C A244420 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).
%C A244420 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).
%C A244420 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, ...
%C A244420 From _Wolfdieter Lang_, Jun 13 2016: (Start)
%C A244420 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).
%C A244420 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.
%C A244420 (End)
%H A244420 Wolfdieter Lang, <a href="/A244420/a244420.pdf">First rows of the triangles.</a>
%F A244420 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.
%F A244420 The rationals R(n, m) = binomial(2*n+1, m)/2^(2*n). - _Wolfdieter Lang_, Jun 12 2016
%e A244420 The numerator triangle a(n,m) begins:
%e A244420   n\m      0      1      2     3    4     5    6   7   8   9
%e A244420   0:       1
%e A244420   1:       3      1
%e A244420   2:       5      5      1
%e A244420   3:      35     21      7     1
%e A244420   4:      63     21      9     9    1
%e A244420   5:     231    165    165    55   11     1
%e A244420   6:     429   1287    715   143   39    13    1
%e A244420   7:    6435   5005   3003  1365  455   105   15   1
%e A244420   8:   12155   2431   1547  1547  595    85   17  17   1
%e A244420   9:   46189  37791  12597  6783 2907   969  969 171  19   1
%e A244420   ...
%e A244420 The rational triangle R(n,m) begins:
%e A244420   n\m       0        1         2       3        4        5
%e A244420   0:        1
%e A244420   1:      3/4      1/4
%e A244420   2:      5/8     5/16      1/16
%e A244420   3:    35/64    21/64      7/64    1/64
%e A244420   4:   63/128    21/64      9/64   9/256    1/256
%e A244420   5:  231/512  165/512  165/1024 55/1024  11/1024   1/1024
%e A244420   ...
%e A244420 The next rows are:
%e A244420   n=6: 429/1024, 1287/4096, 715/4096, 143/2048, 39/2048, 13/4096, 1/4096,
%e A244420   n=7: 6435/16384, 5005/16384, 3003/16384, 1365/16384, 455/16384, 105/16384, 15/16384, 1/16384,
%e A244420   n=8: 12155/32768, 2431/8192, 1547/8192, 1547/16384, 595/16384, 85/8192, 17/8192, 17/65536, 1/65536,
%e A244420   n=9: 46189/131072, 37791/131072, 12597/65536, 6783/65536, 2907/65536, 969/65536, 969/262144, 171/262144, 19/262144, 1/262144,
%e A244420   ...
%e A244420 Expansions:
%e A244420 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).
%e A244420 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.
%e A244420 For the Todd polynomials see the coefficient table A084930.
%Y A244420 Cf. A084930, A244421, A000108, A001790, A046161, A111418, A273496.
%K A244420 nonn,easy,tabl,frac
%O A244420 0,2
%A A244420 _Wolfdieter Lang_, Aug 04 2014