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 A386699 #31 Aug 17 2025 09:53:10
%S A386699 1,7,69,733,8061,90462,1028871,11814376,136643085,1589311381,
%T A386699 18569375114,217773347502,2561944357311,30219704365104,
%U A386699 357278540928168,4232449819704768,50227362114232109,596988743410929087,7105534815529752831,84678089652554263155,1010268312800732117946
%N A386699 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(5*n,k).
%F A386699 a(n) = [x^n] 1/((1-3*x) * (1-x)^(4*n)).
%F A386699 a(n) = Sum_{k=0..n} 3^(n-k) * binomial(4*n+k-1,k).
%F A386699 From _Vaclav Kotesovec_, Jul 30 2025: (Start)
%F A386699 Recurrence: 128*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(3052*n^4 - 15114*n^3 + 26432*n^2 - 18693*n + 4131)*a(n) = 8*(42807352*n^8 - 285737492*n^7 + 758983420*n^6 - 1002945218*n^5 + 644348866*n^4 - 111879380*n^3 - 84004497*n^2 + 44187381*n - 5806080)*a(n-1) - 1215*(5*n - 9)*(5*n - 8)*(5*n - 7)*(5*n - 6)*(3052*n^4 - 2906*n^3 - 598*n^2 + 1037*n - 192)*a(n-2).
%F A386699 a(n) ~ 5^(5*n + 1/2) / (sqrt(Pi*n) * 2^(8*n + 1/2)). (End)
%F A386699 G.f.: g/((3-2*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - _Seiichi Manyama_, Aug 13 2025
%F A386699 a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - _Seiichi Manyama_, Aug 15 2025
%F A386699 G.f.: 1/(1 - x*g^4*(15-8*g)) where g = 1+x*g^5 is the g.f. of A002294. - _Seiichi Manyama_, Aug 17 2025
%t A386699 Table[(243/16)^n - Binomial[5*n, n]*(-1 + Hypergeometric2F1[1, -4*n, 1 + n, -1/2]), {n,0,25}] (* _Vaclav Kotesovec_, Jul 30 2025 *)
%o A386699 (PARI) a(n) = sum(k=0, n, 2^(n-k)*binomial(5*n, k));
%Y A386699 Cf. A000244, A100192, A385004, A385498.
%Y A386699 Cf. A163455, A371739, A386702.
%Y A386699 Cf. A002294.
%K A386699 nonn,easy
%O A386699 0,2
%A A386699 _Seiichi Manyama_, Jul 30 2025