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 A384365 #32 Sep 03 2025 09:10:21
%S A384365 1,9,67,458,2979,18750,115278,696372,4149283,24452534,142808922,
%T A384365 827780684,4767638158,27309438252,155689424316,883891633896,
%U A384365 4999703023395,28188457323366,158463492162594,888473780483292,4969653746436762,27737520941131140,154507945286680452
%N A384365 a(n) = Sum_{k=0..n} (k+1) * 3^k * binomial(2*n+1,n-k).
%H A384365 Vincenzo Librandi, <a href="/A384365/b384365.txt">Table of n, a(n) for n = 0..1000</a>
%F A384365 a(n) = [x^n] 1/((1-4*x)^2 * (1-x)^n).
%F A384365 a(n) = Sum_{k=0..n} 4^k * (-3)^(n-k) * binomial(2*n+1,k) * binomial(2*n-k-1,n-k).
%F A384365 a(n) = Sum_{k=0..n} (k+1) * 4^k * binomial(2*n-k-1,n-k).
%F A384365 G.f.: (1+sqrt(1-4*x))/( 2 * sqrt(1-4*x) * (2*sqrt(1-4*x)-1)^2 ).
%F A384365 D-finite with recurrence +27*n*a(n) +6*(-58*n+17)*a(n-1) +32*(46*n-37)*a(n-2) +1024*(-2*n+3)*a(n-3)=0. - _R. J. Mathar_, Aug 19 2025
%F A384365 a(n) ~ n * 2^(4*n+1) / 3^(n+1). - _Vaclav Kotesovec_, Aug 20 2025
%t A384365 Table[Sum[(k+1) * 3^k*Binomial[2*n+1,n-k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 03 2025 *)
%o A384365 (PARI) a(n) = sum(k=0, n, (k+1)*3^k*binomial(2*n+1, n-k));
%o A384365 (Magma) [&+[ (k+1) * 3^k * Binomial(2*n+1,n-k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 03 2025
%Y A384365 Cf. A100193, A386940, A386941.
%Y A384365 Cf. A088218, A258431, A386955, A386956.
%K A384365 nonn,changed
%O A384365 0,2
%A A384365 _Seiichi Manyama_, Aug 11 2025