A395079 a(n) = Sum_{k=0..n} n^k * binomial(n + 1, k + 1) * binomial(2*n + 2, k) / (n + 1).

1, 3, 33, 667, 19953, 796276, 39884617, 2409056091, 170578159425, 13865428827595, 1273098711647601, 130359650056935828, 14731760791405826353, 1821596921118627413112, 244673069988065590056025, 35478934710659760285365851, 5524449097025217753681737985, 919439893040295438072449964745

Offset

0,2

Comments

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

Links

Paolo Xausa, Table of n, a(n) for n = 0..300

Formula

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

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)).

Mathematica

A395079[n_] := Hypergeometric2F1[-n, -2*(n+1), 2, n];
Array[A395079, 20, 0] (* Paolo Xausa, Apr 12 2026 *)

Prog

(Python)
from math import comb
def A395079(n: int) -> int:
    return sum(((n**i) * comb(n + 1, i + 1) * comb(2 * n + 2, i)) // (n + 1) for i in range(n + 1))
A = [A395079(n) for n in range(20)]; print(A)

Crossrefs

Cf. A395080.

Sequence in context: A364242 A390106 A376390 * A376393 A380666 A379860

Adjacent sequences: A395076 A395077 A395078 * A395080 A395081 A395082

Keyword

nonn

Author

Peter Luschny, Apr 12 2026

Status

approved