A395079 - OEIS

%I #10 Apr 12 2026 16:38:34

%S 1,3,33,667,19953,796276,39884617,2409056091,170578159425,

%T 13865428827595,1273098711647601,130359650056935828,

%U 14731760791405826353,1821596921118627413112,244673069988065590056025,35478934710659760285365851,5524449097025217753681737985,919439893040295438072449964745

%N a(n) = Sum_{k=0..n} n^k * binomial(n + 1, k + 1) * binomial(2*n + 2, k) / (n + 1).

%C The number of ternary trees with n nodes weighted by n colors on the middle and right edges.

%H Paolo Xausa, <a href="/A395079/b395079.txt">Table of n, a(n) for n = 0..300</a>

%F a(n) = A395080(n, n).

%F a(n) ~ e * n^n * C(n + 1) where C denotes the Catalan numbers; also a(n) ~ (4*e / Pi^(1/2)) *((4*n)^n / n^(3/2)).

%t A395079[n_] := Hypergeometric2F1[-n, -2*(n+1), 2, n];

%t Array[A395079, 20, 0] (* _Paolo Xausa_, Apr 12 2026 *)

%o (Python)

%o from math import comb

%o def A395079(n: int) -> int:

%o     return sum(((n**i) * comb(n + 1, i + 1) * comb(2 * n + 2, i)) // (n + 1) for i in range(n + 1))

%o A = [A395079(n) for n in range(20)]; print(A)

%Y Cf. A395080.

%K nonn

%O 0,2

%A _Peter Luschny_, Apr 12 2026