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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. 0
1, 1, 4, 60, 3360, 705600, 558835200, 1678182105600, 19198403288064000, 840083731079104512000, 141100463472046393835520000, 91242050302344912388163665920000, 227753296409896438988240405704212480000, 2199573010737856838816729366169572868096000000, 82356764599728553816070191604819734458909327360000000 (list; graph; refs; listen; history; text; internal format)
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
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
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.
Sequence in context: A132627 A229771 A341275 * A340937 A136385 A173607
KEYWORD
nonn
AUTHOR
Marcel K. Goh, Nov 02 2020
STATUS
approved

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Last modified April 13 17:11 EDT 2024. Contains 371644 sequences. (Running on oeis4.)