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 A378483 #13 Jul 19 2025 11:10:26
%S A378483 1,9,72,549,4077,29772,214884,1537677,10930923,77298849,544300992,
%T A378483 3819184236,26718251868,186440019192,1298115301356,9020928853341,
%U A378483 62582406445287,433509545320731,2998884192348888,20720206275346269,143005275737941437,986000187782876976
%N A378483 Expansion of (Sum_{k>=0} binomial(3*k,k) * x^k)^3.
%F A378483 a(n) = Sum_{i+j+k=n, i,j,k >= 0} binomial(3*i,i) * binomial(3*j,j) * binomial(3*k,k).
%F A378483 G.f.: B(x)^3 where B(x) is the g.f. of A005809.
%F A378483 4*a(n) - 27*a(n-1) = 3*A005809(n) for n > 0.
%F A378483 a(n) ~ sqrt(n) * 3^(3*n + 3/2) / (sqrt(Pi) * 2^(2*n+2)). - _Vaclav Kotesovec_, Jul 19 2025
%t A378483 nmax = 20; CoefficientList[Series[Sum[Binomial[3*k,k] * x^k, {k, 0, nmax}]^3, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 19 2025 *)
%o A378483 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(3*k, k)*x^k)^3)
%Y A378483 Cf. A000302, A378484.
%Y A378483 Cf. A005809.
%K A378483 nonn
%O A378483 0,2
%A A378483 _Seiichi Manyama_, Nov 28 2024