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 A386700 #25 Aug 17 2025 09:56:06
%S A386700 1,0,6,30,186,1140,7116,44856,285066,1823232,11721726,75683718,
%T A386700 490429224,3187723344,20774505408,135699314640,888177411018,
%U A386700 5823660624408,38245666664994,251528316024042,1656338630258826,10919849458481028,72068276593960884,476093333668519872
%N A386700 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(3*n,k).
%F A386700 a(n) = [x^n] 1/((1+2*x) * (1-x)^(2*n)).
%F A386700 a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(2*n+k-1,k).
%F A386700 From _Vaclav Kotesovec_, Jul 30 2025: (Start)
%F A386700 Recurrence: 18*n*(2*n - 1)*(55*n^2 - 175*n + 138)*a(n) = (11605*n^4 - 49410*n^3 + 74243*n^2 - 46014*n + 9720)*a(n-1) + 24*(3*n - 5)*(3*n - 4)*(55*n^2 - 65*n + 18)*a(n-2).
%F A386700 a(n) ~ 3^(3*n + 1/2) / (5 * sqrt(Pi*n) * 2^(2*n)). (End)
%F A386700 G.f.: g/((-2+3*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - _Seiichi Manyama_, Aug 13 2025
%F A386700 a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - _Seiichi Manyama_, Aug 15 2025
%F A386700 G.f.: 1/(1 - 6*x*g^2*(-1+g)) where g = 1+x*g^3 is the g.f. of A001764. - _Seiichi Manyama_, Aug 17 2025
%t A386700 Table[(-8/9)^n - Binomial[3*n, n]*(-1 + Hypergeometric2F1[1, -2*n, 1 + n, 1/3]), {n, 0, 25}] (* _Vaclav Kotesovec_, Jul 30 2025 *)
%o A386700 (PARI) a(n) = sum(k=0, n, (-3)^(n-k)*binomial(3*n, k));
%Y A386700 Cf. A122803, A386701, A386702.
%Y A386700 Cf. A005809, A066380, A165817, A371813, A385004.
%K A386700 nonn,easy
%O A386700 0,3
%A A386700 _Seiichi Manyama_, Jul 30 2025