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%I A122366
%S A122366 1,1,3,1,5,10,1,7,21,35,1,9,36,84,126,1,11,55,165,330,462,1,13,78,286,
%T A122366 715,1287,1716,1,15,105,455,1365,3003,5005,6435,1,17,136,680,2380,6188,
%U A122366 12376,19448,24310,1,19,171,969,3876,11628,27132,50388,75582,92378,1,21
%N A122366 Triangle read by rows: T(n,k) = binomial(2*n+1,k), 0<=k<=n.
%C A122366 T(n,0)=1; for n>0: T(n,1)=n+2; for n>1: T(n,n)=T(n-1,n-2)+3*T(n-1,n-1), 
               T(n,k)=T(n-1,k-2)+2*T(n-1,k-1)+T(n-1,k), 1<k<n.
%C A122366 T(n,k) = A034868(2*n+1,k) = A007318(2*n+1,k), 0<=k<=n;
%C A122366 sum of n-th row = A000302(n) = n^4;
%C A122366 central terms give A052203; T(n,n) = A001700(n).
%C A122366 Reversal of A111418. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 
               22 2007
%C A122366 Coefficient triangle for the expansion of one half of odd powers of 2*x 
               in terms of Chebyshev's T-polynomials: ((2*x)^(2*n+1))/2 = sum(a(n,
               k)*T(2*(n-k)+1,x),k=0..n) with Chebyshev's T-polynomials. See A053120. 
               - W. Lang, Mar 07 2007.
%C A122366 The signed triangle a(n,k)*(-1)^(n-k) appears in the formula (2*sin(phi))^(2*n+1))/
               2 = sum(((-1)^(n-k))*a(n,k)*sin((2*(n-k)+1)*phi),k=0..n) - W. Lang, 
               Mar 07 2007.
%C A122366 The signed triangle a(n,k)*(-1)^(n-k) appears therefore in the formula 
               (4-x^2)^n = sum(((-1)^(n-k))*a(n,k)*S(2*(n-k),x),k=0..n) with Chebyshev's 
               S-polynomials. See A049310 for S(n,x). - W. Lang, Mar 07 2007.
%D A122366 T. J. Rivlin, Chebyshev polynomials: from approximation theory to algebra 
               and number theory, 2. ed., Wiley, New York, 1990. p. 54-5, Ex.1.5.31.
%H A122366 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 A122366 <a href="Sindx_Pas.html#Pascal">Index entries for triangles and arrays 
               related to Pascal's triangle</a>
%H A122366 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/
               Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</
               a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 
               1972, p. 795.
%H A122366 <a href="Sindx_Ch.html#Cheby">Index entries for sequences related to 
               Chebyshev polynomials.</a>
%F A122366 T(n,0)=1; T(n,k)=T(n-1,k-1)*2*n*(2*n+1)/(k*(2*n-k+1)) for k>0.
%e A122366 .......... / 1 \ .......... =A062344(0,0)=A034868(0,0),
%e A122366 ......... / 1 . \ ......... =T(0,0)=A034868(1,0),
%e A122366 ........ / 1 2 . \ ........ =A062344(1,0..1)=A034868(2,0..1),
%e A122366 ....... / 1 3 ... \ ....... =T(1,0..1)=A034868(3,0..1),
%e A122366 ...... / 1 4 6 ... \ ...... =A062344(2,0..2)=A034868(4,0..2),
%e A122366 ..... / 1 5 10 .... \ ..... =T(2,0..2)=A034868(5,0..2),
%e A122366 .... / 1 6 15 20 ... \ .... =A062344(3,0..3)=A034868(6,0..3),
%e A122366 ... / 1 7 21 35 ..... \ ... =T(3,0..3)=A034868(7,0..3),
%e A122366 .. / 1 8 28 56 70 .... \ .. =A062344(4,0..4)=A034868(8,0..4),
%e A122366 . / 1 9 36 84 126 ..... \ . =T(4,0..4)=A034868(9,0..4).
%e A122366 Row n=2:[1,5,10] appears in the expansion ((2*x)^5)/2 = T(5,x)+5*T(3,
               x)+10*T(1,x).
%e A122366 Row n=2:[1,5,10] appears in the expansion ((2*cos(phi))^5)/2 = cos(5*phi)+5*cos(3*phi)+10*cos(1*phi).
%e A122366 The signed row n=2:[1,-5,10] appears in the expansion ((2*sin(*phi))^5)/
               2 = sin(5*phi)-5*sin(3*phi)+10*sin(phi).
%e A122366 The signed row n=2:[1,-5,10] appears therefore in the expansion (4-x^2)^2 
               = S(4,x)-5*S(2,x)+10*S(0,x).
%Y A122366 Cf. A062344.
%Y A122366 Odd numbered rows of A008314. Even numbered rows of A008314 are A127673.
%Y A122366 Sequence in context: A055199 A146916 A146255 this_sequence A103327 A065229 
               A093905
%Y A122366 Adjacent sequences: A122363 A122364 A122365 this_sequence A122367 A122368 
               A122369
%K A122366 nonn,tabl
%O A122366 0,3
%A A122366 Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 30 2006
%E A122366 Chebyshev and trigonometric comments from W. Lang, Mar 07 2007.

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