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 A386701 #25 Aug 17 2025 09:53:23
%S A386701 1,1,13,103,869,7476,65323,577242,5144949,46167196,416527828,
%T A386701 3774785983,34336862435,313330665532,2866982877954,26294890918308,
%U A386701 241665561294741,2225104901535564,20520648006149980,189523353219338572,1752680220372189364,16227703263403842768
%N A386701 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(4*n,k).
%F A386701 a(n) = [x^n] 1/((1+2*x) * (1-x)^(3*n)).
%F A386701 a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(3*n+k-1,k).
%F A386701 From _Vaclav Kotesovec_, Jul 30 2025: (Start)
%F A386701 Recurrence: 81*n*(3*n - 2)*(3*n - 1)*(612*n^3 - 2838*n^2 + 4354*n - 2209)*a(n) = 24*(165240*n^6 - 1019628*n^5 + 2493432*n^4 - 3068178*n^3 + 1984652*n^2 - 632900*n + 76545)*a(n-1) + 128*(2*n - 3)*(4*n - 7)*(4*n - 5)*(612*n^3 - 1002*n^2 + 514*n - 81)*a(n-2).
%F A386701 a(n) ~ 2^(8*n - 1/2) / (sqrt(Pi*n) * 3^(3*n + 1/2)). (End)
%F A386701 G.f.: g/((-2+3*g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Aug 13 2025
%F A386701 a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(4*n,k) * binomial(4*n-k-1,n-k). - _Seiichi Manyama_, Aug 15 2025
%F A386701 G.f.: 1/(1 - x*g^3*(-8+9*g)) where g = 1+x*g^4 is the g.f. of A002293. - _Seiichi Manyama_, Aug 17 2025
%t A386701 Table[(-16/27)^n - Binomial[4*n, n]*(-1 + Hypergeometric2F1[1, -3*n, 1 + n, 1/3]), {n, 0, 25}] (* _Vaclav Kotesovec_, Jul 30 2025 *)
%o A386701 (PARI) a(n) = sum(k=0, n, (-3)^(n-k)*binomial(4*n, k));
%Y A386701 Cf. A386700, A386702.
%Y A386701 Cf. A066381, A262977, A371814, A385498.
%K A386701 nonn,easy
%O A386701 0,3
%A A386701 _Seiichi Manyama_, Jul 30 2025