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A081496
Start with Pascal's triangle; a(n) is the sum of the numbers on the periphery of the n-th central rhombus containing exactly 4 numbers.
3
5, 14, 46, 160, 574, 2100, 7788, 29172, 110110, 418132, 1595620, 6113744, 23505356, 90633800, 350351640, 1357278300, 5268292830, 20483876820, 79765662900, 311038321440, 1214362277700, 4746455801880, 18570960418920, 72728638093800
OFFSET
1,1
LINKS
Qi Wang, Tau-tilting finite simply connected algebras, arXiv:1910.01937 [math.RT], 2019. 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
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
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
AUTHOR
Amarnath Murthy, Mar 25 2003
EXTENSIONS
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004
STATUS
approved