A299443 Expansion of (x^4 + 2*x^3 + 7*x^2 - 6*x + 1)^(-1/2).

1, 3, 10, 35, 127, 474, 1807, 6999, 27436, 108541, 432493, 1733174, 6977777, 28200413, 114338320, 464857475, 1894420045, 7736238420, 31649963275, 129693294945, 532216500532, 2186868151211, 8996351889535, 37048736568870, 152722557174139, 630116066189691

Offset: 0

Author

Peter Luschny, Feb 10 2018

Keywords

• nonn

Examples

			From the first formula follows that a(n) = p_{n}(1) of the polynomials p_{n}(x):
[0]    1
[1]    3
[2]    9 +      x
[3]   27 +    8*x
[4]   81 +   45*x +     x^2
[5]  243 +  216*x +  15*x^2
[6]  729 +  945*x + 132*x^2 +    x^3
[7] 2187 + 3888*x + 900*x^2 + 24*x^3
...
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1602

Crossrefs

Cf. A299444.

Programs

Maple

ogf := (x^4 + 2*x^3 + 7*x^2 - 6*x + 1)^(-1/2):
series(ogf, x, 27): seq(coeff(%,x,n),n=0..25);

Mathematica

CoefficientList[ Series[1/Sqrt[x^4 + 2 x^3 + 7 x^2 - 6 x + 1], {x, 0, 25}], x] (* Robert G. Wilson v, Feb 11 2018 *)

Formula

a(n) = Sum_{k=0..n} 2^k*binomial(n, k)*hypergeom([-k, k - n, k - n], [1, -n], 1/2).

D-finite with recurrence: a(n) = ((2-n)*a(n-4)+(3-2*n)*a(n-3)+(7-7*n)*a(n-2)+(6*n-3)*a(n-1))/n for n >= 4.