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 A335647 #28 Mar 21 2025 05:10:59
%S A335647 1,10,84,715,6188,54264,480700,4292145,38567100,348330136,3159461968,
%T A335647 28760021745,262596783764,2403979904200,22057981462440,
%U A335647 202802465047245,1867897112363100,17231414395464984,159186450151978480,1472474663905800940,13636219405675529520
%N A335647 a(n) = binomial(4*n+1,n+1).
%F A335647 G.f.: A(x) = x*B'(x)/B(x)+x*(1/x-1/B(x))', where B(x) = x*(1+B(x))^4 = A002293(x)-1.
%F A335647 a(n) = Sum_{k=0..n+1} C(n+1,k)*C(3*n,k).
%F A335647 From _Karol A. Penson_, Mar 07 2025: (Start)
%F A335647 G.f.: h(z) = 3*(hypergeom([-1/2,-1/4,1/4],[-2/3,-1/3],256*z/27)-1)/(4*z) satisfies
%F A335647 z^2 + 41*z + 27 + (73*z^2 + 310*z - 27)*h(z) + z*(32*z^2 + 795*z - 81)*h(z)^2 + 3*z^2*(256*z - 27)*h(z)^3 + z^3*(256*z - 27)*h(z)^4 = 0. (End)
%o A335647 (Maxima)
%o A335647 a(n):=sum(binomial(n+1,k)*binomial(3*n,k),k,0,n+1);
%o A335647 (PARI) a(n) = binomial(4*n+1, n+1); \\ _Michel Marcus_, Jun 15 2020
%Y A335647 Cf. A002293.
%K A335647 nonn
%O A335647 0,2
%A A335647 _Vladimir Kruchinin_, Jun 15 2020