A085841 Triangle: row #n has n+1 terms. T(n,m) = 4^m (2n+1)! / ( (2n-2m)! (2m+1)! ).

1, 3, 4, 5, 40, 16, 7, 140, 336, 64, 9, 336, 2016, 2304, 256, 11, 660, 7392, 21120, 14080, 1024, 13, 1144, 20592, 109824, 183040, 79872, 4096, 15, 1820, 48048, 411840, 1281280, 1397760, 430080, 16384

Offset: 0

Author

Gary W. Adamson, Jul 05 2003

Keywords

• nonn
• tabl

Comments

Row #n has the unsigned coefficients of a polynomial whose roots are 2 cot (Pi k / (2n+1)) for k=1..2n.

Polynomial of row #n = Sum_{m=0..n} (-1)^m*T(n,m)*x^(2n-2m).

Examples

			1
3x^2 - 4
5x^4 - 40x^2 + 16
7x^6 - 140x^4 + 336x^2 - 64
9x^8 - 336x^6 + 2016x^4 - 2304x^2 + 256
11x^10 - 660x^8 + 7392x^6 - 21120x^4 + 14080x^2 - 1024
Polynomial #4 has eight roots: 2 cot (Pi k / 9) for k=1..8.
		

Links

• Rui Duarte and António Guedes de Oliveira, A Famous Identity of Hajós in Terms of Sets, Journal of Integer Sequences, Vol. 17 (2014), #14.9.1.

Crossrefs

Cf. A085840.

Programs

PARI

T(n,m) = 4^m*(2*n+1)!/((2*n-2*m)!*(2*m+1)!);
tabl(nn) = for (n=0, nn, for (k=0, n, print1(T(n, k), ", ")); print); \\ Michel Marcus, Mar 18 2018

Extensions

Edited by Don Reble, Nov 13 2005