Search: id:A084534
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%I A084534
%S A084534 1,1,2,1,4,2,1,6,9,2,1,8,20,16,2,1,10,35,50,25,2,1,12,54,112,105,36,2,
               1,
%T A084534 14,77,210,294,196,49,2,1,16,104,352,660,672,336,64,2,1,18,135,546,1287,
%U A084534 1782,1386,540,81,2,1,20,170,800,2275,4004,4290,2640,825,100,2
%N A084534 Triangle read by rows: row #n has n+1 terms. T(n,0)=1, T(n,n)=2, T(n,
               m) = T(n-1,m-1) + sum(k=0 to m) T(n-1-k,m-k).
%C A084534 Sum of row #n = A000204(2n). (But sum of row #0 = 1.)
%C A084534 Row #n has the unsigned coefficients of the monic polynomial whose roots 
               are 2 cos(pi (2k-1)/(4n)) for k=1 to 2n. [Comment corrected by Barry 
               Brent, Jan 03 2006]
%C A084534 The positive roots are some diagonal lengths of a regular (4n)-gon, inscribed 
               in the unit circle.
%C A084534 Polynomial of row #n = sum(m=0 to n) [(-1)^m] T(n,m) x^(2n-2m).
%C A084534 This is the unsigned version of the coefficient table for scaled Chebyshev 
               T(2*n,x) polynomials. - W. Lang, Mar 07 2007
%F A084534 Signed version: a(n,m)=0 if n<m, a(0,0)=1 else a(n,m)=((-1)^m)*binomial(2*n-m,
               m)*2*n/(2*n-m) - W. Lang, Mar 07 2007
%F A084534 Signed version: a(n,m)=0 if n<m, a(0,0)=1 else a(n,m)=((-1)^m)*sum(binomial(m+l,
               l)*binomial(2*n,2*(l+m))/2^(2*(n-m)-1),l=0..n-m.) - W. Lang, Mar 
               07 2007
%F A084534 Signed version: a(n,m)= A127674(n,n-m)/2^(2*(m-n)-1) (scaled coefficients 
               of Chebyshev's T(2*n,x)), decreasing even powers). - W. Lang, Mar 
               07 2007
%e A084534 1
%e A084534 x^2 - 2
%e A084534 x^4 - 4x^2 + 2
%e A084534 x^6 - 6x^4 + 9x^2 - 2
%e A084534 x^8 - 8x^6 + 20x^4 - 16x^2 + 2
%e A084534 x^10 - 10x^8 + 35x^6 - 50x^4 + 25x^2 - 2
%e A084534 Polynomial #4 has 8 roots: 2 sin(pi k/16) for k=1,3,5,...,15.
%Y A084534 Cf. companion triangle A082985.
%Y A084534 Cf. A082985 (unsigned scaled coefficient table for Chebyshev's T(2*n+1, 
               x)polynomials).
%Y A084534 Sequence in context: A158303 A035607 A059370 this_sequence A165899 A104582 
               A133938
%Y A084534 Adjacent sequences: A084531 A084532 A084533 this_sequence A084535 A084536 
               A084537
%K A084534 nonn,tabl
%O A084534 0,3
%A A084534 Gary W. Adamson (qntmpkt(AT)yahoo.com), May 29 2003
%E A084534 Edited by Don Reble (djr(AT)nk.ca), Nov 12 2005
%E A084534 Chebyshev comments and formulae derived from Rivlin reference given in 
               A127674. W. Lang, Mar 07 2007

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