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 A386870 #14 Aug 09 2025 08:28:18
%S A386870 1,14,297,7024,174608,4466622,116403982,3073417652,81935130444,
%T A386870 2200645300312,59455990356377,1614089892481416,43993649464273588,
%U A386870 1203123469832767556,32997093202771098204,907229481990010791100,24997561841045998756604,690088514785377393552360
%N A386870 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+2,k) * binomial(4*n-k,n-k).
%H A386870 Vincenzo Librandi, <a href="/A386870/b386870.txt">Table of n, a(n) for n = 0..350</a>
%F A386870 a(n) = [x^n] (1+x)^(4*n+2)/(1-2*x)^(3*n+1).
%F A386870 a(n) = [x^n] 1/((1-x)^2 * (1-3*x)^(3*n+1)).
%F A386870 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * (n-k+1) * binomial(4*n+2,k).
%F A386870 a(n) = Sum_{k=0..n} 3^k * (n-k+1) * binomial(3*n+k,k).
%t A386870 Table[Sum[2^(n-k)*Binomial[4*n+2,k]*Binomial[4*n-k,n-k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 09 2025 *)
%o A386870 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(4*n+2, k)*binomial(4*n-k, n-k));
%o A386870 (Magma) [&+[2^(n-k)*Binomial(4*n+2,k) * Binomial(4*n-k,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 09 2025
%Y A386870 Cf. A385668, A386845.
%K A386870 nonn
%O A386870 0,2
%A A386870 _Seiichi Manyama_, Aug 06 2025