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 A372002 #12 Apr 22 2024 12:37:41
%S A372002 1,2,0,4,-8,24,-72,224,-720,2368,-7936,27008,-93088,324288,-1140032,
%T A372002 4039296,-14409728,51713792,-186577152,676334592,-2462090752,
%U A372002 8997154816,-32992079872,121362092032,-447721572864,1656081763328,-6140640246784,22820403312640
%N A372002 G.f. A(x) satisfies A(x) = ( 1 + 4*x*(1 + x*A(x)) )^(1/2).
%F A372002 G.f.: A(x) = (1+4*x)/(-2*x^2 + sqrt(1+4*x+4*x^4)).
%F A372002 a(n) = Sum_{k=0..n} 4^k * binomial(n/2-k/2+1/2,k) * binomial(k,n-k)/(n-k+1).
%F A372002 D-finite with recurrence n*a(n) +2*(2*n-3)*a(n-1) +4*(n-6)*a(n-4)=0. - _R. J. Mathar_, Apr 22 2024
%p A372002 A372002 := proc(n)
%p A372002 add(4^k*binomial((n-k+1)/2,k)*binomial(k,n-k)/(n-k+1),k=0..n) ;
%p A372002 end proc:
%p A372002 seq(A372002(n),n=0..60) ; # _R. J. Mathar_, Apr 22 2024
%o A372002 (PARI) my(N=30, x='x+O('x^N)); Vec((1+4*x)/(-2*x^2+sqrt(1+4*x+4*x^4)))
%o A372002 (PARI) a(n) = sum(k=0, n, 4^k*binomial(n/2-k/2+1/2, k)*binomial(k, n-k)/(n-k+1));
%Y A372002 Cf. A372003.
%K A372002 sign
%O A372002 0,2
%A A372002 _Seiichi Manyama_, Apr 15 2024