A081496 - OEIS

%I #14 Mar 04 2022 13:05:42

%S 5,14,46,160,574,2100,7788,29172,110110,418132,1595620,6113744,

%T 23505356,90633800,350351640,1357278300,5268292830,20483876820,

%U 79765662900,311038321440,1214362277700,4746455801880,18570960418920,72728638093800

%N 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.

%H Qi Wang, <a href="https://arxiv.org/abs/1910.01937">Tau-tilting finite simply connected algebras</a>, arXiv:1910.01937 [math.RT], 2019. See Example 5.1 and table page 17.

%F 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

%F 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

%e The first three rhombuses are

%e ...1...........2.........6

%e .1...1.......3...3.....10..10

%e ...2......,....6.....,...20

%e and the corresponding sums are a(1) =5, a(2) =14 and a(3) =46.

%p seq((9*n-4)*binomial(2*(n-1),(n-1))/n,n=1..26); # C. Ronaldo, Dec 20 2004

%o (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

%o {a(n) = sum(k=0,n,A029635(n, k)^2)} \\ _Paul D. Hanna_, Oct 17 2017

%o for(n=1,30,print1(a(n),", "))

%Y Cf. A081494, A081495, A081497, A029635.

%K nonn

%O 1,1

%A _Amarnath Murthy_, Mar 25 2003

%E More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 20 2004