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 A385823 #40 Aug 27 2025 05:19:02
%S A385823 1,1,7,42,256,1586,9949,63004,401930,2579130,16628809,107636402,
%T A385823 699030226,4552602248,29722279084,194458630304,1274628824490,
%U A385823 8368726082346,55027110808177,362301656545966,2388274575638228,15760514137668514,104108685843640517,688331413734386356
%N A385823 a(n) = Sum_{k=0..n} binomial(3*n-3,k).
%H A385823 Vincenzo Librandi, <a href="/A385823/b385823.txt">Table of n, a(n) for n = 0..1000</a>
%F A385823 a(n) = [x^n] (1+x)^(3*n-3)/(1-x).
%F A385823 a(n) = [x^n] 1/((1-x)^(2*n-3) * (1-2*x)).
%F A385823 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-3,k) * binomial(3*n-k-4,n-k).
%F A385823 a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-4,n-k).
%F A385823 G.f.: 1/(g^2 * (2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
%t A385823 Table[Sum[Binomial[3*n-3,k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 27 2025 *)
%o A385823 (PARI) a(n) = sum(k=0, n, binomial(3*n-3, k));
%o A385823 (Magma) [&+[Binomial(3*n-3,k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 27 2025
%Y A385823 Cf. A066380, A160906, A386006, A387007, A387008, A387033.
%Y A385823 Cf. A001764, A047099, A165817.
%K A385823 nonn,changed
%O A385823 0,3
%A A385823 _Seiichi Manyama_, Aug 13 2025