OFFSET
0,2
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..500
FORMULA
a(n) = (2*n + 1) * C(n)^3.
a(n) = (64*n^3 - 32*n^2 - 16*n + 8)*a(n - 1) / (n + 1)^3, for n >= 1.
a(n) = [x^n] hypergeom([1/2, -2*n - 1, -2*n], [2, 2], 4*x) (see A367023). - Peter Luschny, Nov 07 2023
MAPLE
C := n -> binomial(2*n, n)/(n + 1):
# Alternative:
a := proc(n) option remember; if n = 0 then 1 else
(64*n^3 - 32*n^2 - 16*n + 8)*a(n - 1) / (n + 1)^3 fi end: seq(a(n), n = 0..17);
# Third form:
p := n -> hypergeom([1/2, -2*n - 1, -2*n], [2, 2], 4*x):
a := n -> coeff(simplify(p(n)), x, n): seq(a(n), n = 0..17);
MATHEMATICA
Array[(2*#+1)*CatalanNumber[#]^3 &, 20, 0] (* Paolo Xausa, Feb 19 2024 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Nov 16 2022
STATUS
approved