OEIS/Square Root Recurrences

Examples

1 / Sqrt(1 - 2*b*x + d*x^2)

Cf. Noe, equation (4):

A098455: b=2; d=-36; 
RecurrenceTable[{a[0]==1, a[1]==b, n*a[n]==(2*n-1)*b*a[n-1] - (n-1)*d*a[n-2]}, a[n], {n,0,8}]
-> 1, 2, 24, 128, 1096, 7632, 60864, 461568, 3648096
make runholo OFFSET=0 MATRIX="[[0],[36,-36],[2,-4],[0,1]]" INIT="[1,2]"
new HolonomicRecurrence(0, "[[0],[-d,d],[b,-2*b],[0,1]]", "[1,b]", 0);

Order 1

f     = [1/Sqrt[1-b*x],x]]                  a(0) = 1            / 0!
df/dx = b/(2*(1 - b*x)^(3/2))               a(1) = b/2          / 1!
        (3*b^2)/(4*(1 - b*x)^(5/2))         a(2) = 3/4*b        / 2!
        (15*b^3)/(8*(1 - b*x)^(7/2))        a(3) = 3*5/8*b      / 3!
        (105*b^4)/(16*(1 - b*x)^(9/2))      a(4) = 3*5*7/2^4*b  / 4!
=> 2*n*a(n) - b*(2*n-1)*a(n-1) = 0

Order 2

1/(1-b*x-c*x^2)^(1/2)
=> 0! * a(0) = 1
-(-b-2*c*x)/(2*(1-b*x-c*x^2)^(3/2))
=> 1! * a(1) = b/2
(3*(-b-2*c*x)^2)/(4*(1-b*x-c*x^2)^(5/2)) 
    + c/(1-b*x-c*x^2)^(3/2)
=> 2! * a(2) = 3*b^2/4 + c
(-15*(-b-2*c*x)^3)/(8*(1-b*x-c*x^2)^(7/2)) 
    - (9*c*(-b-2*c*x))/(2*(1-b*x-c*x^2)^(5/2))
=> 3! * a(3) = 15/8*b^3 + 9*b*c/2
(105*(-b-2*c*x)^4)/(16*(1-b*x-c*x^2)^(9/2)) 
    + (45*c*(-b-2*c*x)^2)/(2*(1-b*x-c*x^2)^(7/2)) 
    + (9*c^2)/(1-b*x-c*x^2)^(5/2)
=> 4! * a(4) = 105/16*b^4 + 45/2*b^2*c^2 + 9*c^2