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 A387037 #23 Sep 03 2025 03:17:03
%S A387037 1,4,29,232,1941,16664,145499,1285624,11460949,102875128,928495764,
%T A387037 8417689504,76599066579,699232769512,6400175653922,58718827590992,
%U A387037 539822826733397,4971747032359352,45863130731297180,423683961417124576,3919058645835901556
%N A387037 a(n) = Sum_{k=0..n} binomial(4*n-1,k).
%H A387037 Vincenzo Librandi, <a href="/A387037/b387037.txt">Table of n, a(n) for n = 0..1000</a>
%F A387037 a(n) = [x^n] (1+x)^(4*n-1)/(1-x).
%F A387037 a(n) = [x^n] 1/((1-x)^(3*n-1) * (1-2*x)).
%F A387037 a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(4*n-1,k) * binomial(4*n-k-2,n-k).
%F A387037 a(n) = Sum_{k=0..n} 2^k * binomial(4*n-k-2,n-k).
%F A387037 G.f.: 1/((2-g) * (4-3*g)) where g = 1+x*g^4 is the g.f. of A002293.
%F A387037 D-finite with recurrence: 128*(4*n-5)*(4*n-7)*(2*n-3)*(11*n^2-3*n-3)*a(n-2) -8*(946*n^5-4218*n^4+6512*n^3-3753*n^2+201*n+315)*a(n-1) +3*n*(3*n-2)*(3*n-4)*(11*n^2-25*n+11)*a(n) = 0. - _Georg Fischer_, Aug 17 2025
%F A387037 a(n) ~ 2^(8*n - 5/2) / (sqrt(Pi*n) * 3^(3*n - 3/2)). - _Vaclav Kotesovec_, Sep 03 2025
%t A387037 Table[Sum[Binomial[4*n-1,k],{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, Sep 03 2025 *)
%o A387037 (PARI) a(n) = sum(k=0, n, binomial(4*n-1, k));
%o A387037 (Magma) [&+[Binomial(4*n-1, k): k in [0..n]]: n in [0..25]]; // _Vincenzo Librandi_, Sep 03 2025
%Y A387037 Cf. A066381, A386811, A387009, A387010, A387011, A387034, A387035, A387036.
%Y A387037 Cf. A114121, A387033.
%Y A387037 Cf. A002293, A262977.
%K A387037 nonn,changed
%O A387037 0,2
%A A387037 _Seiichi Manyama_, Aug 13 2025