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 A084534 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). 7
 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; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS 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^(2n-2m). This is the unsigned version of the coefficient table for scaled Chebyshev T(2*n,x) polynomials. - Wolfdieter Lang, Mar 07 2007 Reversed A127677 (cf. A156308, A217476, A263916). - Tom Copeland, Nov 07 2015 REFERENCES 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). LINKS I. Kaplansky and J. Riordan, The problème des ménages, Scripta Math. 12, (1946), 113-124. [Scan of annotated copy] FORMULA T(n,m) = binomial(2*n-m, m)*2*n/(2*n-m) for n > 0. - Andrew Howroyd, Dec 18 2017 Signed version: a(n,m)=0 if n 0. Companion triangle A082985. Cf. A082985 (unsigned scaled coefficient table for Chebyshev's T(2*n+1, x) polynomials). Cf. A127677, A156308, A217476, A263916. Sequence in context: A304223 A035607 A059370 * A165899 A316354 A104582 Adjacent sequences:  A084531 A084532 A084533 * A084535 A084536 A084537 KEYWORD nonn,tabl AUTHOR Gary W. Adamson, May 29 2003 EXTENSIONS Edited by Don Reble, Nov 12 2005 STATUS approved

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Last modified June 13 04:49 EDT 2021. Contains 344980 sequences. (Running on oeis4.)