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 A366325 #23 Sep 14 2024 06:53:14
%S A366325 1,2,-1,3,-10,36,-137,543,-2219,9285,-39587,171369,-751236,3328218,
%T A366325 -14878455,67030785,-304036170,1387247580,-6363044315,29323149825,
%U A366325 -135700543190,630375241380,-2938391049395,13739779184085,-64430797069375,302934667061301,-1427763630578197
%N A366325 G.f. satisfies A(x) = (1 + x) * (1 + x/A(x)).
%F A366325 G.f.: A(x) = -2*x*(1+x) / (1+x-sqrt((1+x)*(1+5*x))).
%F A366325 a(n) = (-1)^(n-1) * Sum_{k=0..n} binomial(2*k-1,k) * binomial(n-2,n-k)/(2*k-1).
%F A366325 a(n) ~ -(-1)^n * 5^(n - 1/2) / (2 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Oct 07 2023
%F A366325 From _Peter Bala_, Sep 10 2024: (Start)
%F A366325 a(n) = 1/(1 - n) * Sum_{k = 0..n} binomial(-n+k, k)*binomial(-n+k+1, n-k) for n not equal to 1. Cf. A007863.
%F A366325 a(n) = Sum_{k = 0..n-2} binomial(-n+k+1, k)*binomial(-n+k+1, n-k)/(-n+k+1) for n >= 2.
%F A366325 P-recursive: n*a(n) = - 3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2) with a(1) = 2 and a(2) = -1. (End)
%p A366325 a := proc(n) option remember; if n = 1 then 2 elif n = 2 then -1 else (-3*(2*n - 3)*a(n-1) - 5*(n - 3)*a(n-2))/n fi; end:
%p A366325 seq(a(n), n = 1..30); # _Peter Bala_, Sep 10 2024
%o A366325 (PARI) a(n) = (-1)^(n-1)*sum(k=0, n, binomial(2*k-1, k)*binomial(n-2, n-k)/(2*k-1));
%Y A366325 Cf. A073157, A007863, A364336, A364337, A364338, A364339, A366326, A366327, A366328.
%Y A366325 Cf. A000108, A002212, A112478.
%K A366325 sign
%O A366325 0,2
%A A366325 _Seiichi Manyama_, Oct 07 2023