Search: id:A127675
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%I A127675
%S A127675 1,4,3,16,20,5,64,112,56,7,256,576,432,120,9,1024,2816,2816,1232,220,11,
               4096,
%T A127675 13312,16640,9984,2912,364,13,16384,61440,92160,70400,28800,6048,560,15,
               65536,
%U A127675 278528,487424,452608,239360,71808,11424,816,17,262144,1245184
%V A127675 1,-4,3,16,-20,5,-64,112,-56,7,256,-576,432,-120,9,-1024,2816,-2816,1232,
               -220,11,4096,
%W A127675 -13312,16640,-9984,2912,-364,13,-16384,61440,-92160,70400,-28800,6048,
               -560,15,65536,
%X A127675 -278528,487424,-452608,239360,-71808,11424,-816,17,-262144,1245184
%N A127675 Coefficient table for Chebyshev's U(2*n,x) polynomials in decreasing 
               powers of (1-x^2).
%C A127675 This table gives therefore sin((2*n+1)*phi) in terms of falling odd powers 
               of sin(phi).
%C A127675 The unsigned triangle with reversed rows is A084930 (the signs differ).
%H A127675 W. Lang, <a href="http://www-itp.physik.uni-karlsruhe.de/~wl/EISpub/A127675.text">
               First 15 rows and more. </a>
%F A127675 a(n,m)=0 if n<m else a(n,m)=((-4)^(n-m))*binomial(2n-m,m)*(2*n+1)/(2*(n-m)+1), 
               n>=m>=0. (Proof from the differential eq. for U(2*n,x): (1-x^2)*diff(U(2*n,
               x),x$2) - 3*x*diff(U(2*n,x),x) + 4*n*(n+1)*U(2*n,x)=0.)
%F A127675 a(n,m)=0 if n<m else a(n,m)= sum(binomial(m+k,k)*binomial(2*n+1,2*(m+k))*(-1)^(n-m),
               k=0..n-m) (from de Moivre's formula for sin((2*n+1)*phi) after replacing 
               cos(phi)^2 by 1-sin(phi)^2).
%e A127675 [1];[ -4,3];[16,-20,5];[ -64,112,-56,7];[256,-576,432,-120,9]; ...
%e A127675 Row n=3: -64*(1-x^2)^3+ 112*(1-x^2)^2 -56*(1-x^2)^1 + 7 = 64*x^6 - 80*x^4 
               + 24* x^2 -1 =U(6,x).
%e A127675 Row n=3: sin(7*phi)=-64*sin(phi)^7 + 112*sin(phi)^5 - 56*sin(phi)^3 + 
               7*sin(phi).
%Y A127675 Row sums (signed) A033999(n)=(-1)^n. Row sums (unsigned) A002315(n).
%Y A127675 Cf. A082985 (scaled coefficient table).
%Y A127675 Sequence in context: A065679 A062776 A084471 this_sequence A058557 A038233 
               A046162
%Y A127675 Adjacent sequences: A127672 A127673 A127674 this_sequence A127676 A127677 
               A127678
%K A127675 sign,tabl,easy
%O A127675 0,2
%A A127675 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Mar 07 2007

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