This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A360322 #17 Aug 22 2025 11:02:11
%S A360322 1,2,-4,10,-30,102,-376,1462,-5900,24470,-103644,446382,-1948854,
%T A360322 8605290,-38362200,172423770,-780496110,3554991270,-16281079900,
%U A360322 74927379550,-346328465930,1607078948690,-7483861047480,34963419415650,-163825013554400,769694347677002
%N A360322 a(n) = Sum_{k=0..n} (-5)^(n-k) * binomial(n-1,n-k) * binomial(2*k,k).
%F A360322 G.f.: sqrt( (1+5*x)/(1+x) ).
%F A360322 n*a(n) = 2*(-3*n+4)*a(n-1) - 5*(n-2)*a(n-2).
%F A360322 Sum_{i=0..n} Sum_{j=0..i} (-1/5)^i * a(j) * a(i-j) = (1/5)^n.
%F A360322 a(n) = 2 * (-1)^(n+1) * A007317(n) for n > 0.
%F A360322 From _Seiichi Manyama_, Aug 22 2025: (Start)
%F A360322 a(n) = (-1/4)^n * Sum_{k=0..n} 5^(n-k) * binomial(2*k,k) * binomial(2*(n-k),n-k)/(1-2*(n-k)).
%F A360322 a(n) = (-1)^n * Sum_{k=0..n} binomial(2*k,k)/(1-2*k) * binomial(n-1,n-k). (End)
%o A360322 (PARI) a(n) = sum(k=0, n, (-5)^(n-k)*binomial(n-1, n-k)*binomial(2*k, k));
%o A360322 (PARI) my(N=30, x='x+O('x^N)); Vec(sqrt((1+5*x)/(1+x)))
%Y A360322 Cf. A063886, A085362, A360317, A360318, A360319, A360321.
%Y A360322 Cf. A007317.
%K A360322 sign,easy,changed
%O A360322 0,2
%A A360322 _Seiichi Manyama_, Feb 03 2023