0,3

Sum of row #n = A000204(2n+1).

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

The positive roots are the diagonal lengths of a regular (2n+1)-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 also the unsigned coefficient table of Chebyshev's 2*T(2*n+1,x) polynomials expanded in decreasing odd powers of 2*x. - W. Lang, Mar 07 2007.

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

K. Dilcher and K. B. Stolarsky, A Pascal-type triangle characterizing twin primes, Amer. Math. Monthly, 112 (2005), 673-681.

Stephen Eberhart, "Mathematical-Physical Correspondence," Number 37-38, Jan 08, 1982; Gary W. Adamson, May 29, 2003.

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

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 to 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 (qntmpkt(AT)yahoo.com), May 29 2003

Signed version: a(n,m)=0 if n<m else 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. - W. Lang, Mar 07 2007.

Signed version: a(n,m)=0 if n<m else a(n,m)=((-1)^m)*sum(binomial(m+k,k)*binomial(2*n+1,2*(k+m))/2^(2*(n-m)),k=0..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). - W. Lang, Mar 07 2007.

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). - W. Lang, Mar 07 2007.

Unsigned version: a(n,m)=0 if n<m else a(n,m)=binomial(2*n-m,m)*(2*n+1)/(2*(n-m)+1). From the differential eq. for U(2*n,x). - W. Lang, Mar 07 2007.

1

x^2 - 3

x^4 - 5x^2 + 5

x^6 - 7x^4 + 14x^2 - 7

x^8 - 9x^6 + 27x^4 - 30x^2 + 9

x^10 - 11x^8 + 44x^6 - 77x^4 + 55x^2 - 11

Polynomial #4 has 8 roots: 2 sin(2 pi k/9) for k=1 to 8.

Coefficients give

1

1 3

1 5 5

1 7 14 7

1 9 27 30 9

1 11 44 77 55 11

Cf. companion triangle A084534.

Sequence in context: A131303 A131768 A084533 this_sequence A111125 A072919 A104489

Adjacent sequences: A082982 A082983 A082984 this_sequence A082986 A082987 A082988

nonn,tabl

Gary W. Adamson (qntmpkt(AT)yahoo.com), May 29 2003

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

Re-edited by Don Reble (djr(AT)nk.ca), Nov 12 2005

Chebyshev comments from W. Lang, Mar 07 2007.