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 A371813 #28 Aug 17 2025 11:17:21
%S A371813 1,1,7,40,239,1461,9076,57044,361711,2309467,14827487,95630272,
%T A371813 619111172,4021011580,26187682024,170960159100,1118406332655,
%U A371813 7330011083079,48119501497909,316354663355384,2082573599282359,13726029056757029,90565080767425744
%N A371813 a(n) = Sum_{k=0..n} (-1)^k * binomial(3*n-k-1,n-k).
%F A371813 a(n) = [x^n] 1/((1+x) * (1-x)^(2*n)).
%F A371813 a(n) = binomial(3*n-1, n)*hypergeom([1, -n], [1-3*n], -1). - _Stefano Spezia_, Apr 07 2024
%F A371813 From _Vaclav Kotesovec_, Apr 07 2024: (Start)
%F A371813 Recurrence: 8*n*(2*n - 1)*(28*n^2 - 87*n + 67)*a(n) = 2*(1456*n^4 - 6008*n^3 + 8593*n^2 - 4949*n + 960)*a(n-1) + 3*(3*n - 5)*(3*n - 4)*(28*n^2 - 31*n + 8)*a(n-2).
%F A371813 a(n) ~ 3^(3*n + 1/2) / (sqrt(Pi*n) * 2^(2*n+2)). (End)
%F A371813 a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n,k). - _Seiichi Manyama_, Jul 30 2025
%F A371813 G.f.: g/((-1+2*g) * (3-2*g)) where g = 1+x*g^3 is the g.f. of A001764. - _Seiichi Manyama_, Aug 13 2025
%F A371813 a(n) = Sum_{k=0..n} (-1)^k * 2^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k). - _Seiichi Manyama_, Aug 15 2025
%F A371813 G.f.: 1/(1 - x*g^2*(-3+4*g)) where g = 1+x*g^3 is the g.f. of A001764. - _Seiichi Manyama_, Aug 17 2025
%o A371813 (PARI) a(n) = sum(k=0, n, (-1)^k*binomial(3*n-k-1, n-k));
%Y A371813 Cf. A072547, A371814.
%Y A371813 Cf. A005809, A066380, A165817, A385004, A386700.
%K A371813 nonn
%O A371813 0,3
%A A371813 _Seiichi Manyama_, Apr 06 2024