A082985 Coefficient table for Chebyshev's U(2*n,x) polynomial expanded in decreasing powers of (-4*(1-x^2)).
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, 13, 1, 15, 90, 275, 450, 378, 140, 15, 1, 17, 119, 442, 935, 1122, 714, 204, 17, 1, 19, 152, 665, 1729, 2717, 2508, 1254, 285, 19
Offset: 0
Examples
Expansion of polynomials: x^0; x^2 - 3*x^0; x^4 - 5*x^2 + 5*x^0; x^6 - 7*x^4 + 14*x^2 - 7*x^0; x^8 - 9*x^6 + 27*x^4 - 30*x^2 + 9*x^0; x^10 - 11*x^8 + 44*x^6 - 77*x^4 + 55*x^2 - 11*x^0; ... Polynomial #4 has 8 roots: 2*sin(2*Pi*k/9) for k=1 to 8. Coefficients (with signs removed) are 1; 1, 3; 1, 5, 5; 1, 7, 14, 7; 1, 9, 27, 30, 9; 1, 11, 44, 77, 55, 11; ...
References
- J. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, 1992; see pp. 151,175.
- Stephen Eberhart, "Mathematical-Physical Correspondence," Number 37-38, Jan 08, 1982.
- Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990.
Links
- Vincenzo Librandi, Rows n = 0..100 of triangle, flattened
- K. Dilcher and K. B. Stolarsky, A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112 (2005), 673-681.
- Zhi-Hong Sun, Expansions and identities concerning Lucas sequences, Fibonacci Quart. 44 (2006), no. 2, 145-153. See Theorem 3.1
Crossrefs
Programs
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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
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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
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Maple
A082985 := proc(n,m) binomial(2*n-m,m)*(2*n+1)/(2*n-2*m+1) ; end proc: # R. J. Mathar, Sep 08 2013
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Mathematica
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 *) -
PARI
T(n,k)=binomial(2*n-k,k)*(2*n+1)/(2*n-2*k+1); \\ G. C. Greubel, Dec 30 2019
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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
Formula
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).
T(k, s) = ((2k+1)/(2s+1))*binomial(k+s, 2s), 0 <= s <= k; then transpose the triangle. - Gary W. Adamson, May 29 2003
From Wolfdieter Lang, Mar 07 2007: (Start)
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.
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).
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).
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)
T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-2). - Philippe Deléham, Feb 24 2012
Sum_{i>=0} T(n-i,n-2*i) = A003945(n). - Philippe Deléham, Feb 24 2012
Sum_{i>=0} T(n-i, n-2*i)*4^i = 3^n = A000244(n). - Philippe Deléham, Feb 24 2012
From Paul Weisenhorn Nov 25 2019: (Start)
T(r,k) = binomial(2*r-k,k-1) + binomial(2*r-1-k,k-2) with 1 <= r and 1 <= k <= r.
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)
Extensions
Edited by Anne Donovan (anned3005(AT)aol.com), Jun 11 2003
Re-edited by Don Reble, Nov 12 2005
Comments