OFFSET
0,3
COMMENTS
Determinant of the n X n Hankel matrix whose i-th antidiagonal is filled with the n+i-th Catalan number for i = 0..2*n-2.
[ 5, 14, 42]
a(3) = det( [14, 42, 132] ) = 30.
[42, 132, 429]
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..62
Wikipedia, Counting lattice paths
Wikipedia, Hankel matrix
FORMULA
a(n) = Product_{i=1..n-1, j=i..n-1} (i+j+2*n)/(i+j).
a(n) mod 2 = 1 <=> n in { A131577 }.
a(n) ~ exp(1/24) * 2^(1/6 - n + 8*n^2) / (sqrt(A) * n^(1/24) * 3^(9*n^2/2 - 1/12)), where A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Aug 26 2023
EXAMPLE
a(0) = 1: ( ).
a(1) = 1: (/\).
a(2) = 3: /\ /\ /\
(/\/\, /\/\), (/\/\, / \), (/ \, / \).
G.f. = 1 + x + 3*x^2 + 30*x^3 + 1001*x^4 + 111384*x^5 + 41314284*x^6 + ... - Michael Somos, Jun 27 2023
MAPLE
a:= n-> mul(mul((i+j+2*n)/(i+j), j=i..n-1), i=1..n-1):
seq(a(n), n=0..14);
MATHEMATICA
Join[{1}, Table[Sqrt[2*BarnesG[4*n]] * BarnesG[n] * Gamma[2*n]^(3/2) / BarnesG[3*n + 1], {n, 1, 12}]] (* Vaclav Kotesovec, Aug 26 2023 *)
PROG
(PARI) a(n) = prod(i=1, n-1, prod(j=i, n-1, (i+j+2*n)/(i+j))); \\ Michel Marcus, Jul 05 2022
CROSSREFS
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Jun 30 2022
STATUS
approved