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 A370097 #25 Aug 09 2025 07:33:16
%S A370097 1,5,49,545,6401,77505,956929,11976193,151388161,1928363009,
%T A370097 24712450049,318255628289,4115300220929,53396370030593,
%U A370097 694845537386497,9064787191660545,118516719269445633,1552528215946035201,20372392543502991361,267736366910401413121
%N A370097 a(n) = Sum_{k=0..n} binomial(3*n,k) * binomial(3*n-k-1,n-k).
%F A370097 a(n) = [x^n] ( (1+x)^3/(1-x)^2 )^n.
%F A370097 The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x*(1-x)^2/(1+x)^3 ). See A365842.
%F A370097 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n,k). - _Seiichi Manyama_, Jul 31 2025
%F A370097 a(n) ~ 3^(3*n + 1/2) / (5 * sqrt(Pi*n) * 2^(n-1)). - _Vaclav Kotesovec_, Jul 31 2025
%F A370097 a(n) = Sum_{k=0..n} 2^k * binomial(2*n+k-1,k). - _Seiichi Manyama_, Aug 01 2025
%F A370097 a(n) = [x^n] 1/((1-x) * (1-2*x)^(2*n)). - _Seiichi Manyama_, Aug 09 2025
%t A370097 Table[Sum[2^k*(-1)^(n-k)*Binomial[3*n, k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Jul 31 2025 *)
%o A370097 (PARI) a(n) = sum(k=0, n, binomial(3*n, k)*binomial(3*n-k-1, n-k));
%Y A370097 Cf. A091527, A370098.
%Y A370097 Cf. A119259, A370101.
%Y A370097 Cf. A365842.
%K A370097 nonn
%O A370097 0,2
%A A370097 _Seiichi Manyama_, Feb 10 2024