Search: id:A039991
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%I A039991
%S A039991 1,1,0,2,0,1,4,0,3,0,8,0,8,0,1,16,0,20,0,5,0,32,0,48,0,18,0,1,64,0,112,
%T A039991 0,56,0,7,0,128,0,256,0,160,0,32,0,1,256,0,576,0,432,0,120,0,9,0,512,0,
%U A039991 1280,0,1120,0,400,0,50,0,1,1024,0,2816,0,2816,0,1232,0,220
%V A039991 1,1,0,2,0,-1,4,0,-3,0,8,0,-8,0,1,16,0,-20,0,5,0,32,0,-48,0,18,0,-1,64,
               0,
%W A039991 -112,0,56,0,-7,0,128,0,-256,0,160,0,-32,0,1,256,0,-576,0,432,0,-120,0,
               9,
%X A039991 0,512,0,-1280,0,1120,0,-400,0,50,0,-1,1024,0,-2816,0,2816,0,-1232,0,220
%N A039991 Triangle of coefficients of cos(x)^n in polynomial for cos(nx).
%C A039991 Also triangle of coefficients of Chebyshev polynomials of first kind 
               (T(n,x)) in decreasing order of powers of x.
%D A039991 M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, 
               National Bureau of Standards Applied Math. Series 55, 1964 (and various 
               reprintings), p. 795.
%D A039991 E. A. Guilleman, Synthesis of Passive Networks, Wiley, 1957, p. 593.
%D A039991 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory 
               to algebra and number theory, 2. ed., Wiley, New York, 1990.
%H A039991 T. D. Noe, <a href="b039991.txt">Table of n, a(n) for n=1..100</a>
%H A039991 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.nrbook.com/
               abramowitz_and_stegun/">Handbook of Mathematical Functions</a>, National 
               Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 
               [alternative scanned copy].
%H A039991 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A039991 a(n, m) = 0 if n<m or m odd, (-1)^{m/2} if m=n is even, ((-1)^(3*m/2))*(2^(n-m-1))*n*binomial(n-1-m/
               2, n-1-m)/(n-m) else. a(n, m) = 2*a(n-1, m)-a(n-2, m-2), n >= 2, 
               m >= 0; a(n, -2) := 0=: a(n, -1), a(0, 0)=1=a(1, 0).
%F A039991 G.f. for m-th column: 0 if m odd, (1-x)/(1-2*x) if m=0, else ((-1)^(m/
               2))*(x^m)*(1-x)/(1-2*x)^(m/2+1). For g.f. for row polynomials and 
               row sums, see A053120.
%e A039991 Letting c = cos x, we have: cos 0x = 1, cos 1x = 1c; cos 2x = 2c^2-1; 
               cos 3x = 4c^3-3c, cos 4x = 8c^4-8c^2+1, etc.
%Y A039991 Cf. A028297. Essentially same as A008310.
%Y A039991 Triangle without zeros: A028297. Without signs: A081265.
%Y A039991 Sequence in context: A121298 A121462 A131487 this_sequence A081265 A108643 
               A133838
%Y A039991 Adjacent sequences: A039988 A039989 A039990 this_sequence A039992 A039993 
               A039994
%K A039991 tabl,easy,sign,nice
%O A039991 0,4
%A A039991 Dave Wilson
%E A039991 Entry improved by comments from Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), 
               Jan 11 2000.

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