login
a(n) = Sum_{k=0..n} (-5)^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).
4

%I #14 Feb 04 2023 10:18:58

%S 1,2,-4,10,-30,102,-376,1462,-5900,24470,-103644,446382,-1948854,

%T 8605290,-38362200,172423770,-780496110,3554991270,-16281079900,

%U 74927379550,-346328465930,1607078948690,-7483861047480,34963419415650,-163825013554400,769694347677002

%N a(n) = Sum_{k=0..n} (-5)^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).

%F G.f.: sqrt( (1+5*x)/(1+x) ).

%F n*a(n) = 2*(-3*n+4)*a(n-1) - 5*(n-2)*a(n-2).

%F Sum_{i=0..n} Sum_{j=0..i} (-1/5)^i * a(j) * a(i-j) = (1/5)^n.

%F a(n) = 2 * (-1)^(n+1) * A007317(n) for n > 0.

%o (PARI) a(n) = sum(k=0, n, (-5)^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));

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

%Y Cf. A063886, A085362, A360317, A360318, A360319, A360321.

%Y Cf. A007317.

%K sign,easy

%O 0,2

%A _Seiichi Manyama_, Feb 03 2023