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
Examples
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.
Programs
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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
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)
Extensions
Edited by Don Reble, Nov 13 2005
Comments