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 A386614 #24 Aug 10 2025 08:07:19
%S A386614 0,1,16,220,2880,36850,465536,5834852,72744640,903525715,11191199200,
%T A386614 138323478980,1706860996096,21034268215120,258934785258240,
%U A386614 3184696786012500,39140208951032960,480734044749851305,5901368553964031600,72410017973538837880,888114187330722044800,10888921795007470528060
%N A386614 a(n) = Sum_{k=0..n-1} binomial(5*k+1,k) * binomial(5*n-5*k,n-k-1).
%F A386614 G.f.: g^3 * (g-1)/(5-4*g)^2 where g=1+x*g^5.
%F A386614 G.f.: g/((1-g)^2 * (1-5*g)^2) where g*(1-g)^4 = x.
%F A386614 a(n) = Sum_{k=0..n-1} binomial(5*k+1+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
%F A386614 a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n+2,k).
%F A386614 a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k+2,k).
%F A386614 D-finite with recurrence +35651584*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n) +8192*(56348704*n^4-268019168*n^3+418502324*n^2-264019618*n+57303885)*a(n-1) +160*(-65524820000*n^4+314102050000*n^3-463341186250*n^2+159732814775*n+76118151939)*a(n-2) +62500*(660806875*n^4-1813661250*n^3-5080986250*n^2+20705993100*n-17279228304)*a(n-3) +308935546875*(5*n-11)*(5*n-14)*(5*n-13)*(5*n-17)*a(n-4)=0. - _R. J. Mathar_, Aug 10 2025
%o A386614 (PARI) a(n) = sum(k=0, n-1, binomial(5*k+1, k)*binomial(5*n-5*k, n-k-1));
%Y A386614 Cf. A079678, A386367, A386566, A386613.
%Y A386614 Cf. A006419, A386612, A386616, A386617.
%K A386614 nonn
%O A386614 0,3
%A A386614 _Seiichi Manyama_, Jul 27 2025