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 A386942 #25 Sep 03 2025 05:03:18
%S A386942 1,8,54,340,2060,12180,70812,406656,2313630,13067340,73372728,
%T A386942 410013864,2282066332,12658839200,70017730680,386314361808,
%U A386942 2126818591932,11686657363236,64108376373700,351142219736000,1920711937207140,10493241496749000,57263080117042800
%N A386942 a(n) = Sum_{k=0..n} (2*k+1) * binomial(2*k,k) * binomial(2*n-k,n-k).
%H A386942 Vincenzo Librandi, <a href="/A386942/b386942.txt">Table of n, a(n) for n = 0..1000</a>
%F A386942 a(n) = [x^n] 1/((1-4*x)^(3/2) * (1-x)^(n+1)).
%F A386942 G.f.: 1/sqrt( (1-4*x) * (2*sqrt(1-4*x)-1)^3 ).
%F A386942 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(2*n+3/2,k) * binomial(n-k+1/2,n-k) = Sum_{k=0..n} (2*k+1) * (3/4)^k * binomial(2*k,k) * binomial(2*n+3/2,n-k).
%F A386942 a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+3/2,k) * binomial(2*n-k,n-k).
%t A386942 Table[Sum[(2*k+1) *Binomial[2*k,k]* Binomial[2*n-k,n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 03 2025 *)
%o A386942 (PARI) a(n) = sum(k=0, n, (2*k+1)*binomial(2*k, k)*binomial(2*n-k, n-k));
%o A386942 (Magma) [&+[(2*k+1) * Binomial (2*k, k) *Binomial(2*n-k, n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 03 2025
%Y A386942 Cf. A293490, A377011, A383832.
%Y A386942 Cf. A386940, A386941.
%K A386942 nonn,easy,changed
%O A386942 0,2
%A A386942 _Seiichi Manyama_, Aug 10 2025