%I #36 Sep 21 2025 03:00:53

%S 1,10,84,715,6188,54264,480700,4292145,38567100,348330136,3159461968,

%T 28760021745,262596783764,2403979904200,22057981462440,

%U 202802465047245,1867897112363100,17231414395464984,159186450151978480,1472474663905800940,13636219405675529520

%N a(n) = binomial(4*n+1,n+1).

%F 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 a(n) = Sum_{k=0..n+1} C(n+1,k)*C(3*n,k).

%F From _Karol A. Penson_, Mar 07 2025: (Start)

%F G.f.: h(z) = 3*(hypergeom([-1/2,-1/4,1/4],[-2/3,-1/3],256*z/27)-1)/(4*z).

%F G.f.: h(z) satisfies 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)

%F a(n) ~ 2^(8*n+5/2) / (3^(3*n+1/2) * sqrt(Pi*n)). - _Amiram Eldar_, Sep 21 2025

%t a[n_] := Binomial[4*n+1, n+1]; Array[a, 20, 0] (* _Amiram Eldar_, Sep 21 2025 *)

%o (Maxima)

%o a(n):=sum(binomial(n+1,k)*binomial(3*n,k),k,0,n+1);

%o (PARI) a(n) = binomial(4*n+1, n+1); \\ _Michel Marcus_, Jun 15 2020

%Y Cf. A002293.

%K nonn,easy

%O 0,2

%A _Vladimir Kruchinin_, Jun 15 2020