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 A385639 #35 Aug 07 2025 09:21:48
%S A385639 1,7,69,748,8485,98847,1171884,14066808,170421669,2079531685,
%T A385639 25520363869,314653207128,3894577133356,48362609654548,
%U A385639 602248101550920,7517853111444528,94044248726758821,1178641094940246897,14796230460187072719,186022053254555479500,2341837809478393341885
%N A385639 a(n) = Sum_{k=0..n} binomial(4*n+1,k) * binomial(2*n-k,n-k).
%F A385639 a(n) = [x^n] (1+x)^(4*n+1)/(1-x)^(n+1).
%F A385639 a(n) = [x^n] 1/((1-x)^(2*n+1) * (1-2*x)^(n+1)).
%F A385639 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n+1,k) * binomial(3*n-k,n-k).
%F A385639 a(n) = Sum_{k=0..n} 2^k * binomial(n+k,k) * binomial(3*n-k,n-k).
%F A385639 a(n) = binomial(2*n, n)*hypergeom([-1-4*n, -n], [-2*n], -1). - _Stefano Spezia_, Aug 07 2025
%F A385639 a(n) ~ sqrt((187 - 3*sqrt(17)) / (17*Pi*n)) * (51*sqrt(17) - 107)^n / 2^(3*n + 3/2). - _Vaclav Kotesovec_, Aug 07 2025
%t A385639 Table[Sum[Binomial[4*n+1, k]*Binomial[2*n-k, n-k], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 07 2025 *)
%o A385639 (PARI) a(n) = sum(k=0, n, binomial(4*n+1, k)*binomial(2*n-k, n-k));
%Y A385639 Cf. A098430, A383716, A386811.
%Y A385639 Cf. A178792, A244038, A386895.
%K A385639 nonn
%O A385639 0,2
%A A385639 _Seiichi Manyama_, Aug 07 2025