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 A383832 #40 Sep 03 2025 04:24:10
%S A383832 1,10,78,548,3630,23148,143724,874888,5245038,31065500,182189348,
%T A383832 1059775608,6122246572,35160205752,200902089240,1142857957392,
%U A383832 6475994731758,36569545322364,205869970843764,1155749458070040,6472151016349284,36161680227612456,201628061114911848
%N A383832 a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(2*n+2,n-k).
%H A383832 Vincenzo Librandi, <a href="/A383832/b383832.txt">Table of n, a(n) for n = 0..1000</a>
%F A383832 a(n) = [x^n] 1/((1-4*x)^2 * (1-x)^(n+1)).
%F A383832 a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+2,k) * binomial(2*n-k,n-k).
%F A383832 a(n) = Sum_{k=0..n} (k+1) * 4^k * binomial(2*n-k,n-k).
%F A383832 G.f.: 1/( sqrt(1-4*x) * (2*sqrt(1-4*x)-1)^2 ).
%F A383832 D-finite with recurrence 15*n*a(n) +2*(-94*n+23)*a(n-1) +192*(4*n-3)*a(n-2) +512*(-2*n+3)*a(n-3)=0. - _R. J. Mathar_, Aug 21 2025
%t A383832 Table[Sum[(k+1)* 3^k * Binomial[2*n+2,n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 03 2025 *)
%o A383832 (PARI) a(n) = sum(k=0, n, (k+1)*3^k*binomial(2*n+2, n-k));
%o A383832 (Magma) [&+[(k+1) * 3^k * Binomial(2*n+2,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 03 2025
%Y A383832 Cf. A293490, A377011, A386942.
%K A383832 nonn,changed
%O A383832 0,2
%A A383832 _Seiichi Manyama_, Aug 11 2025