A299443 - OEIS

%I #14 Jan 30 2020 21:29:18

%S 1,3,10,35,127,474,1807,6999,27436,108541,432493,1733174,6977777,

%T 28200413,114338320,464857475,1894420045,7736238420,31649963275,

%U 129693294945,532216500532,2186868151211,8996351889535,37048736568870,152722557174139,630116066189691

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

%H Robert Israel, <a href="/A299443/b299443.txt">Table of n, a(n) for n = 0..1602</a>

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

%F 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.

%e From the first formula follows that a(n) = p_{n}(1) of the polynomials p_{n}(x):

%e [0]    1

%e [1]    3

%e [2]    9 +      x

%e [3]   27 +    8*x

%e [4]   81 +   45*x +     x^2

%e [5]  243 +  216*x +  15*x^2

%e [6]  729 +  945*x + 132*x^2 +    x^3

%e [7] 2187 + 3888*x + 900*x^2 + 24*x^3

%e ...

%p ogf := (x^4 + 2*x^3 + 7*x^2 - 6*x + 1)^(-1/2):

%p series(ogf, x, 27): seq(coeff(%,x,n),n=0..25);

%t 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 *)

%Y Cf. A299444.

%K nonn

%O 0,2

%A _Peter Luschny_, Feb 10 2018