A108554 Main diagonal of square array A108553, in which row n equals the crystal ball sequence for D_n lattice.

1, 3, 13, 147, 1681, 18733, 204205, 2195399, 23429185, 249258777, 2650436845, 28209301211, 300745906449, 3212758605189, 34392151363021, 368908132170511, 3964622452867329, 42682164804034609, 460242445806795853

Offset: 0

Author

Paul D. Hanna, Jun 10 2005

Keywords

• nonn
• changed

Comments

Limit a(n+1)/a(n) ~ 11.08...

Limit is equal to A001622^5 = A244593 = 11.090169943749474241... - Vaclav Kotesovec, Aug 31 2025

Crossrefs

Cf. A108553, A108555, A244593.

Programs

Mathematica

Join[{1, 3}, Table[Sum[Binomial[2*n - j, n - j]*(Binomial[2*n, 2*j] - 2*j*(n - j)*Binomial[n, j]/(n - 1)), {j, 0, n}], {n, 2, 20}]] (* Vaclav Kotesovec, Aug 31 2025 *)

PARI

{a(n)=if(n<0,0,if(n==0,1,if(n==1,3,sum(j=0,n,binomial(2*n-j,n-j)* (binomial(2*n,2*j)-2*j*(n-j)*binomial(n,j)/(n-1))))))}

Formula

a(n) ~ phi^(5*n + 1) / (2*5^(1/4)*sqrt(Pi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Aug 31 2025