A386387 a(n) = Sum_{k=0..n} 4^k * binomial(n,k) * Catalan(k).

1, 5, 41, 429, 5073, 64469, 859385, 11853949, 167763361, 2422342053, 35543185353, 528450589005, 7943934373233, 120537517728117, 1843702988611737, 28397640862311453, 440070304667718465, 6856488470912854853, 107340528355762710377, 1687682549936270584045

Offset: 0

Author

Seiichi Manyama, Aug 20 2025

Keywords

• nonn
• changed

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..600

Crossrefs

Column k=4 of A340968.

Cf. A386389, A007317.

Programs

Magma

[&+[4^k*Binomial(n,k) * Catalan(k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, Aug 27 2025

Mathematica

Table[Sum[4^k*Binomial[n,k]*CatalanNumber[k],{k,0,n}],{n,0,25}] (* Vincenzo Librandi, Aug 27 2025 *)
A386387[n_] := Hypergeometric2F1[1/2, -n, 2, -16]; Table[A386387[n], {n, 0, 19}]  (* Peter Luschny, Aug 27 2025 *)

PARI

a(n) = sum(k=0, n, 4^k*binomial(n, k)*(2*k)!/(k!*(k+1)!));

Formula

G.f.: 2/(1 - x + sqrt((1-x) * (1-17*x))).

G.f. A(x) satisfies A(x) = 1/(1 - x) + 4*x*A(x)^2.

a(n) = 1 + 4 * Sum_{k=0..n-1} a(k) * a(n-1-k).

(n+1)*a(n) = (18*n-8)*a(n-1) - 17*(n-1)*a(n-2) for n > 1.

a(n) ~ 17^(n + 3/2) / (64*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 20 2025

a(n) = hypergeom([1/2, -n], [2], -16). - Peter Luschny, Aug 27 2025