%I #68 Sep 08 2022 08:45:10

%S 1,1,3,1,5,5,1,7,14,7,1,9,27,30,9,1,11,44,77,55,11,1,13,65,156,182,91,

%T 13,1,15,90,275,450,378,140,15,1,17,119,442,935,1122,714,204,17,1,19,

%U 152,665,1729,2717,2508,1254,285,19

%N Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).

%C Sum of row #n = A000204(2n+1), i.e., A002878(n).

%C Row #n has the unsigned coefficients of a polynomial whose roots are 2 sin(2*Pi*k/(2n+1)) [for k=1 to 2n].

%C The positive roots are the diagonal lengths of a regular (2n+1)-gon, inscribed in the unit circle.

%C Polynomial of row #n = Sum_{m=0..n} (-1)^m T(n,m) x^(2n-2m).

%C This is also the unsigned coefficient table of Chebyshev's 2*T(2*n+1,x) polynomials expanded in decreasing odd powers of 2*x. - _Wolfdieter Lang_, Mar 07 2007

%C The n-th row are the coefficients of the polynomial S(n) where S(0)=1, S(1)=x+3, and S(n) = (x+2)*S(n-1) - S(n-2) (see Sun link). - _Michel Marcus_, Mar 07 2016

%D J. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, 1992; see pp. 151,175.

%D Stephen Eberhart, "Mathematical-Physical Correspondence," Number 37-38, Jan 08, 1982.

%D Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.

%H Vincenzo Librandi, <a href="/A082985/b082985.txt">Rows n = 0..100 of triangle, flattened</a>

%H K. Dilcher and K. B. Stolarsky, <a href="http://www.maa.org/programs/maa-awards/writing-awards/a-pascal-type-triangle-characterizing-twin-primes">A Pascal-type triangle characterizing twin primes</a>, Amer. Math. Monthly, 112 (2005), 673-681.

%H Zhi-Hong Sun, <a href="http://www.fq.math.ca/Papers1/44-2/quartzhihong02_2006.pdf">Expansions and identities concerning Lucas sequences</a>, Fibonacci Quart. 44 (2006), no. 2, 145-153. See Theorem 3.1

%F Triangle read by rows: row #n has n+1 terms. T(n,0)=1, T(n,n)=2n+1, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k, m-k).

%F T(k, s) = ((2k+1)/(2s+1))*binomial(k+s, 2s), 0 <= s <= k; then transpose the triangle. - _Gary W. Adamson_, May 29 2003

%F From _Wolfdieter Lang_, Mar 07 2007: (Start)

%F Signed version: a(n,m)=0 if n < m, otherwise a(n,m) = ((-1)^m)*binomial(2*n+1-m,m)*(2*n+1)/(2*n+1-m). From the Rivlin reference, p. 37, eq.(1.92), using the differential eq. for T(2*n+1,x). Also from Waring's formula.

%F Signed version: a(n,m)=0 if n < m, otherwise a(n,m) = ((-1)^m)*(Sum_{k=0..n-m} binomial(m+k,k)*binomial(2*n+1,2*(k+m))/2^(2*(n-m)). Proof: De Moivre's formula for cos((2*n+1)*phi) rewritten in terms of odd powers of cos(phi). Cf. Rivlin reference p. 4, eq.(1.10).

%F Signed version: a(n,m) = A084930(n,n-m)/2^(2*(n-m)) (scaled coefficients of Chebyshev's T(2*n+1,x)), decreasing odd powers).

%F Unsigned version: a(n,m)=0 if n < m, otherwise a(n,m) = binomial(2*n-m,m)*(2*n+1)/(2*(n-m)+1). From the differential eq. for U(2*n,x). (End)

%F T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2). - _Philippe Deléham_, Feb 24 2012

%F Sum_{i>=0} T(n-i,n-2*i) = A003945(n). - _Philippe Deléham_, Feb 24 2012

%F Sum_{i>=0} T(n-i, n-2*i)*4^i = 3^n = A000244(n). - _Philippe Deléham_, Feb 24 2012

%F From _Paul Weisenhorn_ Nov 25 2019: (Start)

%F T(r,k) = binomial(2*r-k,k-1) + binomial(2*r-1-k,k-2) with 1 <= r and 1 <= k <= r.

%F For a given n, one gets r = floor((1+sqrt(8*n))/2), k = n-(r^2-r)/2, a(n) = binomial(2*r-k,k-1) + binomial(2*r-1-k,k-2). (End)

%e Expansion of polynomials:

%e x^0;

%e x^2 - 3*x^0;

%e x^4 - 5*x^2 + 5*x^0;

%e x^6 - 7*x^4 + 14*x^2 - 7*x^0;

%e x^8 - 9*x^6 + 27*x^4 - 30*x^2 + 9*x^0;

%e x^10 - 11*x^8 + 44*x^6 - 77*x^4 + 55*x^2 - 11*x^0; ...

%e Polynomial #4 has 8 roots: 2*sin(2*Pi*k/9) for k=1 to 8.

%e Coefficients (with signs removed) are

%e 1;

%e 1, 3;

%e 1, 5, 5;

%e 1, 7, 14, 7;

%e 1, 9, 27, 30, 9;

%e 1, 11, 44, 77, 55, 11;

%e ...

%p A082985 := proc(n,m)

%p binomial(2*n-m,m)*(2*n+1)/(2*n-2*m+1) ;

%p end proc: # _R. J. Mathar_, Sep 08 2013

%t T[n_, m_]:= Binomial[2*n-m, m]*(2*n+1)/(2*n-2*m+1); Table[T[n, m], {n, 0, 9}, {m, 0, n}]//Flatten (* _Jean-François Alcover_, Oct 08 2013, after _R. J. Mathar_ *)

%o (PARI) T(n,k)=binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1); \\ _G. C. Greubel_, Dec 30 2019

%o (Magma) [Binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1): k in [0..n], n in [0..10]]; // _G. C. Greubel_, Dec 30 2019

%o (Sage) [[binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1) for k in (0..n)] for n in (0..10)] # _G. C. Greubel_, Dec 30 2019

%o (GAP) Flat(List([0..10], n-> List([0..n], k-> Binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1) ))); # _G. C. Greubel_, Dec 30 2019

%Y Cf. companion triangle A084534.

%Y Cf. diagonals: A005408, A000330, A005585, A050486, A054333, A057788.

%Y Cf. A054142, A172431.

%K nonn,easy,tabl

%O 0,3

%A _Gary W. Adamson_, May 29 2003

%E Edited by Anne Donovan (anned3005(AT)aol.com), Jun 11 2003

%E Re-edited by _Don Reble_, Nov 12 2005