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Expansion of ((1 - 4*x)/sqrt(1 - 8*x + 4*x^2) - 1)/(6*x^2).
2

%I #24 May 05 2021 01:54:02

%S 1,8,57,400,2810,19824,140497,999968,7143966,51206320,368094122,

%T 2652720096,19159794004,138658606688,1005231020865,7299082678336,

%U 53074479789878,386419850997552,2816685368479342,20553133273532000,150120362670452076

%N Expansion of ((1 - 4*x)/sqrt(1 - 8*x + 4*x^2) - 1)/(6*x^2).

%H Seiichi Manyama, <a href="/A331792/b331792.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = (2/(n+2)) * A331515(n) = Sum_{k=0..n} 3^k * binomial(n+1,k) * binomial(n+1,k+1).

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

%F a(n) ~ 2^(n + 1/2) * (2 + sqrt(3))^(n + 3/2) / (3^(3/4) * sqrt(Pi*n)). - _Vaclav Kotesovec_, Jan 26 2020

%t a[n_] := Sum[3^k * Binomial[n + 1, k] * Binomial[n + 1, k + 1], {k, 0, n}]; Array[a, 21, 0] (* _Amiram Eldar_, May 05 2021 *)

%o (PARI) N=20; x='x+O('x^N); Vec(((1-4*x)/sqrt(1-8*x+4*x^2)-1)/(6*x^2))

%o (PARI) {a(n) = sum(k=0, n, 3^k*binomial(n+1, k)*binomial(n+1, k+1))}

%Y Column 4 of A331791.

%Y Cf. A331515.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Jan 26 2020