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Expansion of 1/sqrt(1 - 4*x*(1+x)^3).
6

%I #20 Jul 12 2024 16:40:35

%S 1,2,12,62,342,1932,11094,64480,378150,2233304,13263772,79136844,

%T 473969586,2847911596,17159547804,103640073972,627280131594,

%U 3803643145596,23102172930156,140522319418164,855880464524472,5219168576004184,31861229045809436

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

%H Seiichi Manyama, <a href="/A361812/b361812.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} 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.

%F a(n) = binomial(2*n, n)*hypergeom([(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4], [1/3-n, 1/2-n, 2/3-n], -2^6/3^3). - _Stefano Spezia_, Jul 11 2024

%t a[n_]:=Binomial[2*n, n]HypergeometricPFQ[{(1-3*n)/4, (2-3*n)/4, 3*(1-n)/4, -3*n/4}, {1/3-n, 1/2-n, 2/3-n}, -2^6/3^3]; Array[a,23,0] (* _Stefano Spezia_, Jul 11 2024 *)

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

%Y Column k=3 of A361830.

%Y Cf. A006139, A137635, A360133, A361790, A361791, A361792, A361813, A361814.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Mar 25 2023