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 A374497 #21 Aug 20 2025 10:33:30
%S A374497 1,6,36,200,1080,5712,29792,153792,787680,4009280,20304768,102405888,
%T A374497 514678528,2579028480,12890311680,64283809792,319954540032,
%U A374497 1589720712192,7886437652480,39069462835200,193307835764736,955361266917376,4716674314223616,23264437702656000
%N A374497 Expansion of 1/(1 - 4*x - 4*x^2)^(3/2).
%F A374497 a(0) = 1, a(1) = 6; a(n) = (2*(2*n+1)*a(n-1) + 4*(n+1)*a(n-2))/n.
%F A374497 a(n) = binomial(n+2,2) * A071356(n).
%F A374497 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(k,n-k). - _Seiichi Manyama_, Oct 19 2024
%F A374497 a(n) = ((n+2)/2) * Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+1,n-2*k) * binomial(2*k+1,k). - _Seiichi Manyama_, Aug 20 2025
%t A374497 a[n_]:= Sum[(2*k+1)*Binomial[2*k,k]*Binomial[k,n-k],{k,0,n}]; Array[a,24,0] (* _Stefano Spezia_, May 08 2025 *)
%o A374497 (PARI) a(n) = binomial(n+2, 2)*sum(k=0, n\2, 2^(n-k)*binomial(n, 2*k)*binomial(2*k, k)/(k+1));
%Y A374497 Cf. A001788, A002457, A025163.
%Y A374497 Cf. A006139, A374511, A374513.
%Y A374497 Cf. A071356.
%K A374497 nonn
%O A374497 0,2
%A A374497 _Seiichi Manyama_, Jul 09 2024