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 A387033 #20 Aug 27 2025 14:27:07
%S A387033 1,3,16,93,562,3473,21778,137980,880970,5658537,36519556,236618693,
%T A387033 1538132224,10026362492,65513177704,428957009288,2813768603466,
%U A387033 18486790962201,121634649321208,801330506737399,5285305708097522,34896814868837161,230631268849574378
%N A387033 a(n) = Sum_{k=0..n} binomial(3*n-1,k).
%H A387033 Vincenzo Librandi, <a href="/A387033/b387033.txt">Table of n, a(n) for n = 0..1000</a>
%F A387033 a(n) = [x^n] (1+x)^(3*n-1)/(1-x).
%F A387033 a(n) = [x^n] 1/((1-x)^(2*n-1) * (1-2*x)).
%F A387033 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n-1,k) * binomial(3*n-k-2,n-k).
%F A387033 a(n) = Sum_{k=0..n} 2^k * binomial(3*n-k-2,n-k).
%F A387033 G.f.: 1/((2-g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764.
%F A387033 D-finite with recurrence: 24*(5*n+6)*(3*n-4)*(3*n-5)*a(n-2)-(295*n^3-451*n^2-234*n+360)*a(n-1)+2*n*(5*n+1)*(2*n-3)*a(n) = 0. - _Georg Fischer_, Aug 17 2025
%F A387033 a(n) ~ 3^(3*n - 1/2) / (sqrt(Pi*n) * 2^(2*n-1)). - _Vaclav Kotesovec_, Aug 27 2025
%t A387033 Table[Sum[Binomial[3*n-1,k],{k,0,n}],{n,0,25}] (* _Vincenzo Librandi_, Aug 27 2025 *)
%o A387033 (PARI) a(n) = sum(k=0, n, binomial(3*n-1, k));
%o A387033 (Magma) [&+[Binomial(3*n-1,k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Aug 27 2025
%Y A387033 Cf. A066380, A160906, A385823, A386006, A387007, A387008.
%Y A387033 Cf. A114121, A387037.
%Y A387033 Cf. A001764, A047099, A165817.
%K A387033 nonn,changed
%O A387033 0,2
%A A387033 _Seiichi Manyama_, Aug 13 2025