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Expansion of 1/sqrt(1 - 4*x/(1-x)^5).
9

%I #24 Mar 28 2023 14:00:50

%S 1,2,16,110,770,5512,40066,294484,2182850,16288430,122198926,

%T 920820578,6964483628,52840433000,401990254180,3065365241440,

%U 23422905551018,179302895759782,1374785979255880,10556280995419090,81161958814162700,624750086745027388

%N Expansion of 1/sqrt(1 - 4*x/(1-x)^5).

%H Winston de Greef, <a href="/A359758/b359758.txt">Table of n, a(n) for n = 0..1110</a>

%F a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(n+4*k-1,n-k).

%F n*a(n) = (10*n-8)*a(n-1) - (19*n-46)*a(n-2) + 20*(n-3)*a(n-3) - 15*(n-4)*a(n-4) + 6*(n-5)*a(n-5) - (n-6)*a(n-6) for n > 5.

%F a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n+k) * binomial(n+3-k,4) * a(k).

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x)^5))

%o (PARI) a(n)=sum(k=0, n, binomial(2*k,k) * binomial(n+4*k-1,n-k)) \\ _Winston de Greef_, Mar 24 2023

%Y Cf. A085362, A110170, A162478, A359489, A360132.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 24 2023