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Expansion of 1/sqrt(1 - 4*x - 8*x^3).
3

%I #13 Jul 13 2024 13:48:11

%S 1,2,6,24,94,372,1508,6192,25638,106908,448356,1889040,7989676,

%T 33902504,144259944,615330784,2630199942,11263613484,48315367076,

%U 207556060816,892819376964,3845161246424,16578320962104,71548426931616,309070048163676,1336223562436632

%N Expansion of 1/sqrt(1 - 4*x - 8*x^3).

%H Seiichi Manyama, <a href="/A374598/b374598.txt">Table of n, a(n) for n = 0..1000</a>

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

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

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x-8*x^3))

%o (PARI) a(n) = sum(k=0, n\3, 2^k*binomial(n-2*k, k)*binomial(2*(n-2*k), n-2*k));

%Y Cf. A157004, A360266.

%Y Cf. A084609, A374599.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jul 13 2024