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Search: id:A084534
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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 to m) T(n-1-k,m-k). |
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+0
5
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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
(list; table; graph; listen)
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OFFSET
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0,3
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COMMENT
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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 to 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 to n) [(-1)^m] T(n,m) x^(2n-2m).
This is the unsigned version of the coefficient table for scaled Chebyshev T(2*n,x) polynomials. - W. Lang, Mar 07 2007
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FORMULA
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Signed version: 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) - W. Lang, Mar 07 2007
Signed version: a(n,m)=0 if n<m, a(0,0)=1 else a(n,m)=((-1)^m)*sum(binomial(m+l,l)*binomial(2*n,2*(l+m))/2^(2*(n-m)-1),l=0..n-m.) - W. Lang, Mar 07 2007
Signed version: a(n,m)= A127674(n,n-m)/2^(2*(m-n)-1) (scaled coefficients of Chebyshev's T(2*n,x)), decreasing even powers). - W. Lang, Mar 07 2007
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EXAMPLE
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1
x^2 - 2
x^4 - 4x^2 + 2
x^6 - 6x^4 + 9x^2 - 2
x^8 - 8x^6 + 20x^4 - 16x^2 + 2
x^10 - 10x^8 + 35x^6 - 50x^4 + 25x^2 - 2
Polynomial #4 has 8 roots: 2 sin(pi k/16) for k=1,3,5,...,15.
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CROSSREFS
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Cf. companion triangle A082985.
Cf. A082985 (unsigned scaled coefficient table for Chebyshev's T(2*n+1, x)polynomials).
Sequence in context: A158303 A035607 A059370 this_sequence A165899 A104582 A133938
Adjacent sequences: A084531 A084532 A084533 this_sequence A084535 A084536 A084537
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KEYWORD
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nonn,tabl
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AUTHOR
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Gary W. Adamson (qntmpkt(AT)yahoo.com), May 29 2003
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EXTENSIONS
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Edited by Don Reble (djr(AT)nk.ca), Nov 12 2005
Chebyshev comments and formulae derived from Rivlin reference given in A127674. W. Lang, Mar 07 2007
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