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%I A049310
%S A049310 1,0,1,1,0,1,0,2,0,1,1,0,3,0,1,0,3,0,4,0,1,1,0,6,0,5,0,1,0,4,0,
%T A049310 10,0,6,0,1,1,0,10,0,15,0,7,0,1,0,5,0,20,0,21,0,8,0,1,1,0,15,0,
%U A049310 35,0,28,0,9,0,1,0,6,0,35,0,56,0,36,0,10,0,1,1,0,21,0,70,0,84,0
%V A049310 1,0,1,-1,0,1,0,-2,0,1,1,0,-3,0,1,0,3,0,-4,0,1,-1,0,6,0,-5,0,1,0,-4,0,
%W A049310 10,0,-6,0,1,1,0,-10,0,15,0,-7,0,1,0,5,0,-20,0,21,0,-8,0,1,-1,0,15,0,
%X A049310 -35,0,28,0,-9,0,1,0,-6,0,35,0,-56,0,36,0,-10,0,1,1,0,-21,0,70,0,-84,0
%N A049310 Triangle of coefficients of Chebyshev's S(n,x) := U(n,x/2) polynomials 
               (exponents in increasing order).
%C A049310 G.f. for row polynomials S(n,x) (signed triangle): 1/(1-x*z+z^2). Unsigned 
               triangle |a(n,m)| has Fibonacci polynomials F(n+1,x) as row polynomials 
               with G.f. 1/(1-x*z-z^2). |a(n,m)| triangle has rows of Pascal's triangle 
               A007318 in the even numbered diagonals (odd numbered ones have only 
               0's).
%C A049310 Row sums (unsigned triangle) A000045(n+1) (Fibonacci). Row sums (signed 
               triangle) S(n,1) sequence = periodic(1,1,0,-1,-1,0) = A010892.
%C A049310 S(n,x) is the characteristic polynomial of the adjacency matrix of the 
               n-path. - Michael Somos, Jun 24 2002
%C A049310 |T(n,k)|=number of compositions of n+1 into k+1 odd parts. Example: |T(7,
               3)|=10 because we have (1,1,3,3),(1,3,1,3),(1,3,3,1),(3,1,1,3),(3,
               1,3,1),(3,3,1,1), (1,1,1,5),(1,1,5,1),(1,5,1,1) and (5,1,1,1). - 
               Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 09 2005
%D A049310 D. S. Mitrinovic, Analytic Inequalities, Springer-Verlag, 1970; p. 232, 
               Sect. 3.3.38.
%D A049310 Theodore J. Rivlin, Chebyshev polynomials: from approximation theory 
               to algebra and number theory, 2. ed., Wiley, New York, 1990.
%H A049310 T. D. Noe, <a href="b049310.txt">Rows 0 to 100 of the triangle, flattened.</
               a>
%H A049310 S. R. Finch, P. Sebah and Z.-Q. Bai, <a href="http://arXiv.org/abs/0802.2654">
               Odd Entries in Pascal's Trinomial Triangle</a> (arXiv:0802.2654)
%H A049310 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FibonacciPolynomial.html">Fibonacci Polynomial</a>
%H A049310 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A049310 T(n, k) := 0 if n<k or n+k odd, else ((-1)^((n+k)/2+k))*binomial((n+k)/
               2, k); T(n, k) = -T(n-2, k)+T(n-1, k-1), T(n, -1) := 0 =: T(-1, k), 
               T(0, 0)=1, T(n, k)= 0 if n<k or n+k odd; G.f. k-th column: (1/(1+x^2)^(k+1))*x^k. 
               - Michael Somos, Jun 24 2002
%F A049310 T(n, k)=binomial((n+k)/2, (n-k)/2)*cos(pi*(n-k)/2)*(1+(-1)^(n-k))/2; 
               - Paul Barry (pbarry(AT)wit.ie), Aug 28 2005
%F A049310 Sum_{k=0..n} T(n, k)^2 = A051286(n) . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 21 2005
%F A049310 Recurrence for the (unsigned) Fibonacci polynomials: F[1]=1, F[2]=x; 
               for n>2, F[n] = x*F[n-1]+F[n-2].
%e A049310 {1}; {0,1}; {-1,0,1}; {0,-2,0,1}; {1,0,-3,0,1};... E.g. fourth row {0,
               -2,0,1} corresponds to polynomial S(3,x)= -2*x+x^3.
%e A049310 Triangle of absolute values of coefficients (coefficients of Fibonacci 
               polynomials) with exponents in increasing order begins:
%e A049310 [1]
%e A049310 [0, 1]
%e A049310 [1, 0, 1]
%e A049310 [0, 2, 0, 1]
%e A049310 [1, 0, 3, 0, 1]
%e A049310 [0, 3, 0, 4, 0, 1]
%e A049310 [1, 0, 6, 0, 5, 0, 1]
%e A049310 [0, 4, 0, 10, 0, 6, 0, 1]
%e A049310 [1, 0, 10, 0, 15, 0, 7, 0, 1]
%e A049310 [0, 5, 0, 20, 0, 21, 0, 8, 0, 1]
%o A049310 (PARI) T(n,k)=if(k<0|k>n|(n+k)%2,0,(-1)^((n+k)/2+k)*binomial((n+k)/2,
               k))
%Y A049310 Cf. A010892.
%Y A049310 Reflection of A053119. Without zeros: A053112.
%Y A049310 Sequence in context: A029407 A099544 A036414 this_sequence A036851 A036850 
               A113206
%Y A049310 Adjacent sequences: A049307 A049308 A049309 this_sequence A049311 A049312 
               A049313
%K A049310 easy,nice,sign,tabl,core
%O A049310 0,8
%A A049310 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de)

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