Search: id:A053120
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%I A053120
%S A053120 1,0,1,1,0,2,0,3,0,4,1,0,8,0,8,0,5,0,20,0,16,1,0,18,0,48,0,32,0,7,0,56,
%T A053120 0,112,0,64,1,0,32,0,160,0,256,0,128,0,9,0,120,0,432,0,576,0,256,1,0,
%U A053120 50,0,400,0,1120,0,1280,0,512,0,11,0,220,0,1232,0,2816,0,2816
%V A053120 1,0,1,-1,0,2,0,-3,0,4,1,0,-8,0,8,0,5,0,-20,0,16,-1,0,18,0,-48,
%W A053120 0,32,0,-7,0,56,0,-112,0,64,1,0,-32,0,160,0,-256,0,128,0,9,0,
%X A053120 -120,0,432,0,-576,0,256,-1,0,50,0,-400,0,1120,0,-1280,0,512,0,-11,0,220,
               0,-1232,0,2816,0,-2816
%N A053120 Triangle of coefficients of Chebyshev's T(n,x) polynomials (powers of 
               x in increasing order).
%C A053120 a(n,m) = A039991(n,n-m).
%C A053120 G.f. for row polynomials T(n,x) (signed triangle): (1-x*z)/(1-2*x*z+z^2). 
               If unsigned:(1-x*z)/(1-2*x*z-z^2).
%C A053120 Row sums (signed triangle): A000012 (powers of 1). Row sums (unsigned 
               triangle): A001333(n).
%D A053120 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory 
               to algebra and number theory, 2. ed., Wiley, New York, 1990.
%H A053120 T. D. Noe, <a href="b053120.txt">Rows 0 to 100 of triangle, flattened</
               a>
%H A053120 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A053120.text">
               Rows n=0..20</a>
%H A053120 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A053120 a(n, m) := 0 if n<m or n+m odd; a(n, m)= (-1)^n/2 if m=0 (n even); else 
               a(n, m)=((-1)^((n+m)/ 2+m))*(2^(m-1))*n*binomial((n+m)/2-1, m-1)/
               m.
%F A053120 Recursion for n >= 2: a(n, m) = 2*a(n-1, m-1)-a(n-2, m), a(n, m)=0 if 
               n<m, a(n, -1) := 0, a(0, 0)=1=a(1, 1).
%F A053120 G.f. for m-th column (signed triangle): 1/(1+x^2) if m=0 else (2^(m-1))*(x^m)*(1-x^2)/
               (1+x^2)^(m+1).
%e A053120 {1}; {0,1}; {-1,0,2}; {0,-3,0,4}; {1,0,-8,0,8};... E.g. fourth row (n=3) 
               corresponds to polynomial T(3,x)= -3*x+4*x^3.
%o A053120 (MAGMA) &cat[ Coefficients(ChebyshevT(n)): n in [0..11] ]; - Klaus Brockhaus 
               (klaus-brockhaus(AT)t-online.de), Mar 08 2008
%Y A053120 Cf. A039991, A000012, A001333.
%Y A053120 Sequence in context: A073739 A046767 A115720 this_sequence A008743 A029179 
               A008721
%Y A053120 Adjacent sequences: A053117 A053118 A053119 this_sequence A053121 A053122 
               A053123
%K A053120 easy,nice,sign,tabl
%O A053120 0,6
%A A053120 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

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