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 A378504 #20 Jul 24 2025 04:04:03
%S A378504 1,12,114,984,8055,63744,492702,3742704,28053423,208057260,1529802648,
%T A378504 11168142048,81041199876,585045970992,4204705925670,30101448952032,
%U A378504 214756404746031,1527491122906212,10834911076417458,76666402505673720,541277205506059743
%N A378504 Expansion of (Sum_{k>=0} binomial(3*k,k) * x^k)^4.
%H A378504 Marko Riedel et al, <a href="https://math.stackexchange.com/questions/5084451/">Identity between two generating functions</a>
%F A378504 a(n) = Sum_{i+j+k+l=n, i,j,k,l >= 0} binomial(3*i,i) * binomial(3*j,j) * binomial(3*k,k) * binomial(3*l,l).
%F A378504 G.f.: B(x)^4 where B(x) is the g.f. of A005809.
%F A378504 4*a(n) - 27*a(n-1) = 3*A006256(n) + A005809(n) for n > 0.
%F A378504 Sum_{n >= 0} a(n) * z^n / (1+z)^(3*n) = (1+z)^4 / (1-2*z)^4. - _Marko Riedel_, Jul 22 2025
%F A378504 From _Vaclav Kotesovec_, Jul 23 2025: (Start)
%F A378504 Recurrence: 8*(n-1)*n*(2*n - 1)*a(n) = 6*(n-1)*(36*n^2 - 9*n - 5)*a(n-1) - 81*n*(3*n - 2)*(3*n - 1)*a(n-2).
%F A378504 a(n) ~ n * 3^(3*n+2) / 2^(2*n+4). (End)
%t A378504 nmax = 20; CoefficientList[Series[Sum[Binomial[3*k, k] * x^k, {k, 0, nmax}]^4, {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jul 23 2025 *)
%o A378504 (PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, binomial(3*k, k)*x^k)^4)
%Y A378504 Cf. A005809, A006256, A378483.
%Y A378504 Cf. A002697, A378484.
%K A378504 nonn
%O A378504 0,2
%A A378504 _Seiichi Manyama_, Nov 28 2024