OFFSET
1,1
LINKS
Qi Wang, Tau-tilting finite simply connected algebras, arXiv:1910.01937 [math.RT], 2019-2022. See Example 5.1 and table page 17.
FORMULA
a(n) = (9*n-4)*Catalan(n-1) = (9*n-4)*binomial(2*(n-1), (n-1))/n. - C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004
a(n) = Sum_{k=0..n} A029635(n,k)^2 for n>=1, where A029635 is the Lucas triangle. - Paul D. Hanna, Oct 17 2017
a(n) ~ 9 * 4^(n-1) / sqrt(Pi*n). - Amiram Eldar, Oct 04 2025
EXAMPLE
The first three rhombuses are
...1...........2.........6
.1...1.......3...3.....10..10
...2......,....6.....,...20
and the corresponding sums are a(1) = 5, a(2) = 14 and a(3) = 46.
MAPLE
seq((9*n-4)*binomial(2*(n-1), (n-1))/n, n=1..26); # C. Ronaldo, Dec 20 2004
MATHEMATICA
a[n_] := (9*n-4) * CatalanNumber[n-1]; Array[a, 30] (* Amiram Eldar, Oct 04 2025 *)
PROG
(PARI) { A029635(n, k) = if( k<0 || k>n, 0, (n==0) + binomial(n, k) + binomial(n-1, k-1))}; \\ program from Michael Somos in A029635
{a(n) = sum(k=0, n, A029635(n, k)^2)} \\ Paul D. Hanna, Oct 17 2017
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amarnath Murthy, Mar 25 2003
EXTENSIONS
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004
STATUS
approved