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Expansion of 1/sqrt(1 - 4*x*(1-x)^3).
5

%I #12 Mar 28 2023 10:48:26

%S 1,2,0,-10,-22,12,174,344,-354,-3304,-5780,9180,65258,99132,-226620,

%T -1313580,-1690990,5441340,26681700,28070100,-128211552,-543818824,

%U -440381780,2978145240,11080939914,6162798092,-68377892976,-225107280388,-64286124152

%N Expansion of 1/sqrt(1 - 4*x*(1-x)^3).

%H Seiichi Manyama, <a href="/A361816/b361816.txt">Table of n, a(n) for n = 0..1000</a>

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

%F n*a(n) = 2 * ( (2*n-1)*a(n-1) - 3*(2*n-2)*a(n-2) + 3*(2*n-3)*a(n-3) - (2*n-4)*a(n-4) ) for n > 3.

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x*(1-x)^3))

%Y Column k=3 of A361834.

%Y Cf. A085362, A110170, A162478, A359489, A359758, A360132, A361815, A361817.

%Y Cf. A361812.

%K sign

%O 0,2

%A _Seiichi Manyama_, Mar 25 2023