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 A377204 #42 May 11 2025 11:47:46
%S A377204 1,0,6,6,36,66,236,546,1626,4106,11388,29646,79838,209718,557328,
%T A377204 1465970,3869448,10166370,26726080,70092570,183756378,481048010,
%U A377204 1258494768,3289100958,8590288128,22418099982,58467588768,152388145382,396954437202,1033452111702,2689186662552
%N A377204 Expansion of 1/(1 - 4*x^2/(1-x))^(3/2).
%H A377204 Vincenzo Librandi, <a href="/A377204/b377204.txt">Table of n, a(n) for n = 0..1500</a>
%F A377204 a(n) = (2*(n-1)*a(n-1) + (3*n+6)*a(n-2) - 2*(2*n-3)*a(n-3))/n for n > 2.
%F A377204 a(n) = Sum_{k=0..floor(n/2)} (2*k+1) * binomial(2*k,k) * binomial(n-k-1,n-2*k).
%F A377204 a(n) ~ sqrt(n) * 2^(3*n - 1/2) / (17^(3/4) * sqrt(Pi) * (sqrt(17) - 1)^(n - 3/2)). - _Vaclav Kotesovec_, May 03 2025
%t A377204 Table[Sum[(2*k+1)*Binomial[2*k,k]*Binomial[n-k-1,n-2*k],{k,0,Floor[n/2]}],{n,0,35}] (* _Vincenzo Librandi_, May 08 2025 *)
%o A377204 (PARI) a(n) = sum(k=0, n\2, (2*k+1)*binomial(2*k, k)*binomial(n-k-1, n-2*k));
%o A377204 (Magma) R<x>:=PowerSeriesRing(Rationals(), 35); Coefficients(R!( 1/(1 - 4*x^2/(1-x))^(3/2))); // _Vincenzo Librandi_, May 08 2025
%Y A377204 Cf. A377197, A377213.
%Y A377204 Cf. A026585, A377215.
%Y A377204 Cf. A377186.
%K A377204 nonn
%O A377204 0,3
%A A377204 _Seiichi Manyama_, Oct 20 2024