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A084534
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Triangle read by rows: row #n has n+1 terms. T(n,0)=1, T(n,n)=2, T(n,m) = T(n-1,m-1) + Sum_{k=0..m} T(n-1-k,m-k).
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8
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1, 1, 2, 1, 4, 2, 1, 6, 9, 2, 1, 8, 20, 16, 2, 1, 10, 35, 50, 25, 2, 1, 12, 54, 112, 105, 36, 2, 1, 14, 77, 210, 294, 196, 49, 2, 1, 16, 104, 352, 660, 672, 336, 64, 2, 1, 18, 135, 546, 1287, 1782, 1386, 540, 81, 2, 1, 20, 170, 800, 2275, 4004, 4290, 2640, 825, 100, 2
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OFFSET
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0,3
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COMMENTS
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Sum of row #n = A000204(2n). (But sum of row #0 = 1.)
Row #n has the unsigned coefficients of the monic polynomial whose roots are 2 cos(Pi*(2k-1)/(4n)) for k=1..2n. [Comment corrected by Barry Brent, Jan 03 2006]
The positive roots are some diagonal lengths of a regular (4n)-gon, inscribed in the unit circle.
Polynomial of row #n = Sum_{m=0..n} (-1)^m * T(n,m) x^(2*n-2*m).
This is the unsigned version of the coefficient table for scaled Chebyshev T(2*n,x) polynomials. - Wolfdieter Lang, Mar 07 2007
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REFERENCES
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I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. See p. 118.
Theodore J. Rivlin, Chebyshev polynomials: from approximation theory to algebra and number theory, 2. ed., Wiley, New York, 1990. p. 37, eq.(1.96) and p. 4. eq.(1.10).
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LINKS
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FORMULA
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T(n,m) = binomial(2*n-m, m)*2*n/(2*n-m) for n > 0. - Andrew Howroyd, Dec 18 2017
a(n,m)=0 if n<m, a(0,0)=1 else a(n,m) = (-1)^m*binomial(2*n-m, m)*2*n/(2*n-m).
a(n,m)=0 if n<m, a(0,0)=1 else a(n,m) = (-1)^m*Sum_{l=0..n-m} binomial(m+l,l)*binomial(2*n,2*(l+m))/2^(2*(n-m)-1).
a(n,m)=0 if n<m, a(0,0)=1 else a(n,m) = A127674(n,n-m)/2^(2*(n-m)-1) (scaled coefficients of Chebyshev's T(2*n,x)), decreasing even powers). [corrected by Johannes W. Meijer, May 31 2018] (End)
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EXAMPLE
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First few Chebyshev T(2*n,x) polynomials:
T(2*0,x) = 1;
T(2*1,x) = x^2 - 2;
T(2*2,x) = x^4 - 4*x^2 + 2;
T(2*3,x) = x^6 - 6*x^4 + 9*x^2 - 2;
T(2*4,x) = x^8 - 8*x^6 + 20*x^4 - 16*x^2 + 2;
T(2*5,x) = x^10 - 10*x^8 + 35*x^6 - 50*x^4 + 25*x^2 - 2;
Triangle begins as:
1;
1, 2;
1, 4, 2;
1, 6, 9, 2;
1, 8, 20, 16, 2;
1, 10, 35, 50, 25, 2;
1, 12, 54, 112, 105, 36, 2;
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MAPLE
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T := proc(n, m): if n=0 then 1 else binomial(2*n-m, m)*2*n/(2*n-m) fi: end: seq(seq(T(n, m), m=0..n), n=0..10); # Johannes W. Meijer, May 31 2018
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MATHEMATICA
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a[n_, m_] := Binomial[2n-m, m]*2n/(2n-m); a[0, 0] = 1; Table[a[n, m], {n, 0, 10}, {m, 0, n}] // Flatten (* Jean-François Alcover, Apr 12 2016, after Wolfdieter Lang *)
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PROG
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(PARI) T(n, m) = if(n==0, m==0, binomial(2*n-m, m)*2*n/(2*n-m)) \\ Andrew Howroyd, Dec 18 2017
(Magma)
A084534:= func< n, k | k eq 0 select 1 else 2*(n/k)*Binomial(2*n-k-1, k-1) >;
(Sage)
def A084534(n, k): return 1 if (k==0) else 2*(n/k)*binomial(2*n-k-1, k-1)
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CROSSREFS
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Cf. A082985 (unsigned scaled coefficient table for Chebyshev's T(2*n+1, x) polynomials).
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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