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 A386702 #25 Aug 17 2025 09:53:39
%S A386702 1,2,24,248,2676,29562,331956,3771896,43242660,499215146,5795429764,
%T A386702 67587697872,791232339756,9292673328174,109440405341088,
%U A386702 1291977861163968,15284200451058724,181147979395807002,2150493166839159936,25567085678133719880,304368033788893315896
%N A386702 a(n) = Sum_{k=0..n} (-3)^(n-k) * binomial(5*n,k).
%F A386702 a(n) = [x^n] 1/((1+2*x) * (1-x)^(4*n)).
%F A386702 a(n) = Sum_{k=0..n} (-2)^(n-k) * binomial(4*n+k-1,k).
%F A386702 From _Vaclav Kotesovec_, Jul 30 2025: (Start)
%F A386702 Recurrence: 648*n*(2*n - 1)*(4*n - 3)*(4*n - 1)*(37331*n^4 - 227972*n^3 + 518701*n^2 - 521044*n + 194928)*a(n) = (9143593823*n^8 - 74277961298*n^7 + 253684378280*n^6 - 473415527402*n^5 + 524935655069*n^4 - 351762123620*n^3 + 137998180332*n^2 - 28677229776*n + 2380855680)*a(n-1) + 160*(5*n - 9)*(5*n - 8)*(5*n - 7)*(5*n - 6)*(37331*n^4 - 78648*n^3 + 58771*n^2 - 18234*n + 1944)*a(n-2).
%F A386702 a(n) ~ 5^(5*n + 1/2) / (7 * sqrt(Pi*n) * 2^(8*n - 1/2)). (End)
%F A386702 G.f.: g/((-2+3*g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - _Seiichi Manyama_, Aug 13 2025
%F A386702 a(n) = Sum_{k=0..n} (-2)^k * 3^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - _Seiichi Manyama_, Aug 15 2025
%F A386702 G.f.: 1/(1 - x*g^4*(-10+12*g)) where g = 1+x*g^5 is the g.f. of A002294. - _Seiichi Manyama_, Aug 17 2025
%t A386702 Table[(-32/81)^n - Binomial[5*n, n]*(-1 + Hypergeometric2F1[1, -4*n, 1 + n, 1/3]), {n, 0, 25}] (* _Vaclav Kotesovec_, Jul 30 2025 *)
%o A386702 (PARI) a(n) = sum(k=0, n, (-3)^(n-k)*binomial(5*n, k));
%Y A386702 Cf. A386700, A386701.
%Y A386702 Cf. A163455, A371739, A386699.
%K A386702 nonn,easy
%O A386702 0,2
%A A386702 _Seiichi Manyama_, Jul 30 2025