A338550 Number of binary trees of height n such that the number of nodes at depth d equals d+1 for every d = 0..n.

1, 1, 4, 60, 3360, 705600, 558835200, 1678182105600, 19198403288064000, 840083731079104512000, 141100463472046393835520000, 91242050302344912388163665920000, 227753296409896438988240405704212480000, 2199573010737856838816729366169572868096000000, 82356764599728553816070191604819734458909327360000000

Offset

0,3

Comments

To satisfy the constraint, there must be n+1 nodes at depth n, and there are 2n allowed slots for a new node.

A binary tree with such a level profile contains A000217(n+1) nodes.

Links

Table of n, a(n) for n=0..14.

Formula

a(n) = binomial(2*n,n+1)*a(n-1), a(0)=1.

a(n) = Product_{k=1..n} binomial(2*k,k+1).

a(n) = 2^(n^2+n-1/24)*A^(3/2)*Pi^(-1/4-n/2)*G(3/2 + n)*Gamma(1 + n)/(exp(1/8)*G(3 + n)) where A is the Glaisher-Kinkelin constant and G is the Barnes G function. - Stefano Spezia, Nov 02 2020

a(n) ~ A^(3/2) * 2^(-7/24 + n + n^2) * exp(-1/8 + n/2) / (n^(11/8 + n/2) * Pi^((n+1)/2)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 29 2023

a(n) = Product_{1 <= j <= i <= n-1} (i + j + 2)/(i - j + 1). - Peter Bala, Oct 25 2024

Mathematica

Table[Product[Binomial[2k, k+1], {k, n}], {n, 0, 14}] (* or *)
Table[2^(n^2+n-1/24)Glaisher^(3/2)Pi^(-1/4-n/2)BarnesG[3/2+n]Gamma[1+n]/(Exp[1/8]BarnesG[3+n]), {n, 0, 14}] (* Stefano Spezia, Nov 02 2020 *)

Crossrefs

Cf. A000217, A007685, A203471, A296589.

Sequence in context: A132627 A229771 A341275 * A340937 A136385 A173607

Adjacent sequences: A338547 A338548 A338549 * A338551 A338552 A338553

Keyword

nonn,easy

Author

Marcel K. Goh, Nov 02 2020

Status

approved