OFFSET
0,2
COMMENTS
a(0) = 1 by convention.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: 1+(8*x+1)*2*x/(1-4*x)^(5/2).
a(n) = C(2*n,n)*C(2*n,2) for n>0, a(0)=1.
a(n) = 2*C(2*(n-1),n-1)*(2*n-1)^2, a(0)=1.
a(n) = 2*(2*n-1)^2*a(n-1)/((n-1)*(2*n-3)) for n>1, a(n) = 2^n for n=0..1.
a(n) = A245687(2n,n).
Sum_{n>=0} 1/a(n) = 1 - log(sqrt(3)+2)*Pi/6 + 4*G/3, where G is Catalan's constant (A006752). - Amiram Eldar, Mar 12 2023
EXAMPLE
a(1) = 2: 12, 21.
a(2) = 36: 1122, 1133, 1144, 1212, 1221, 1313, 1331, 1414, 1441, 2112, 2121, 2211, 2233, 2244, 2323, 2332, 2424, 2442, 3113, 3131, 3223, 3232, 3311, 3322, 3344, 3434, 3443, 4114, 4141, 4224, 4242, 4334, 4343, 4411, 4422, 4433.
MAPLE
a:= proc(n) option remember; `if`(n<2, 2^n,
2*(2*n-1)^2*a(n-1)/((n-1)*(2*n-3)))
end:
seq(a(n), n=0..30);
MATHEMATICA
a[n_] := (2*n^3 + n^2 - n) * CatalanNumber[n]; a[0] = 1; Array[a, 30, 0] (* Amiram Eldar, Mar 12 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Alois P. Heinz, May 20 2016
STATUS
approved