A085840 Triangle read by rows: T(n,m) = 4^m * (2*n+1)! / ( (2*n - 2*m + 1)! * (2*m)! ), row n has n+1 terms.

1, 1, 12, 1, 40, 80, 1, 84, 560, 448, 1, 144, 2016, 5376, 2304, 1, 220, 5280, 29568, 42240, 11264, 1, 312, 11440, 109824, 329472, 292864, 53248, 1, 420, 21840, 320320, 1647360, 3075072, 1863680, 245760

Offset

0,3

Comments

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

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

Links

Table of n, a(n) for n=0..35.

Formula

From Peter Bala, Apr 10 2017: (Start)

O.g.f.: (1 - (1 - 4*x)*t)/(1 - 2*(1 + 4*x)*t + (1 - 4*x)^2*t^2) = 1 + (1 + 12*x)*t + (1 + 40*x + 80*x^2)*t^2 + ....

n_th row polynomial R(n,x) = 1/2*( (1 + 2*sqrt(x))^(2*n+1) + (1 - 2*sqrt(x))^(2*n+1) ). These polynomials occur in the expansion cosh((2*n + 1)*arctanh(2*x)) = R(n,x^2)/(1 - 4*x^2)^(n+1/2). See A285043 - A285046.

For n >= 1, R(n,x) = (1 - 4*x)^n( U(n,(1 + 4*x)/(1 - 4*x)) - U(n-1,(1 + 4*x)/(1 - 4*x)) ), where U(n,x) is the n-th Chebyshev polynomial of the second kind. (End)

Example

1
x^2 - 12
x^4 - 40x^2 + 80
x^6 - 84x^4 + 560x^2 - 448
x^8 - 144x^6 + 2016x^4 - 5376x^2 + 2304
x^10 - 220x^8 + 5280x^6 - 29568x^4 + 42240x^2 - 11264
Polynomial #4 has eight roots: 2 tan (Pi*k/9) for k=1 to 8.

Maple

for n from 0 to 10 do lprint(seq(4^k*binomial(2*n + 1, 2*k), k = 0..n)) end do; # Peter Bala, Apr 10 2017

Crossrefs

Cf. A085841, A285043, A284044, A285045, A285046.

Sequence in context: A036185 A013619 A092527 * A276966 A277799 A278468

Adjacent sequences: A085837 A085838 A085839 * A085841 A085842 A085843

Keyword

nonn,tabl,easy

Author

Gary W. Adamson, Jul 05 2003

Extensions

Edited by Don Reble, Nov 13 2005

Status

approved