This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A338550 #23 Oct 25 2024 14:26:01
%S A338550 1,1,4,60,3360,705600,558835200,1678182105600,19198403288064000,
%T A338550 840083731079104512000,141100463472046393835520000,
%U A338550 91242050302344912388163665920000,227753296409896438988240405704212480000,2199573010737856838816729366169572868096000000,82356764599728553816070191604819734458909327360000000
%N 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.
%C A338550 To satisfy the constraint, there must be n+1 nodes at depth n, and there are 2n allowed slots for a new node.
%C A338550 A binary tree with such a level profile contains A000217(n+1) nodes.
%F A338550 a(n) = binomial(2*n,n+1)*a(n-1), a(0)=1.
%F A338550 a(n) = Product_{k=1..n} binomial(2*k,k+1).
%F A338550 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
%F A338550 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
%F A338550 a(n) = Product_{1 <= j <= i <= n-1} (i + j + 2)/(i - j + 1). - _Peter Bala_, Oct 25 2024
%t A338550 Table[Product[Binomial[2k,k+1],{k,n}],{n,0,14}] (* or *)
%t A338550 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 *)
%Y A338550 Cf. A000217, A007685, A203471, A296589.
%K A338550 nonn,easy
%O A338550 0,3
%A A338550 _Marcel K. Goh_, Nov 02 2020