A108999 Main diagonal of square array A108998, in which row n equals the coordination sequence of B_n lattice.

1, 2, 16, 170, 1856, 20082, 214864, 2282394, 24165120, 255708578, 2708805776, 28752157898, 305908697152, 3262741154194, 34882914424528, 373781033269306, 4013444615232512, 43174945822078530, 465247083731404048

Offset: 0

Author

Paul D. Hanna, Jun 17 2005

Keywords

• nonn
• changed

Comments

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

Links

• Muniru A Asiru, Table of n, a(n) for n = 0..900

Crossrefs

Cf. A108998, A109000, A109001.

Programs

GAP

List([0..20],n->Sum([0..n],j->Binomial(2*n-j-1,n-j)*(Binomial(2*n+1,2*j)-2*n*Binomial(n-1,j-1)))); # Muniru A Asiru, Nov 21 2018

Mathematica

a[n_]:= Sum[Binomial[2*n-j-1, n-j]*(Binomial[2*n+1, 2*j] - 2*n*Binomial[n-1, j-1]), {j,0,n}]; Array[a, 20, 0] (* Stefano Spezia, Nov 21 2018 *)

PARI

{a(n)=sum(j=0,n, binomial(2*n-j-1,n-j)*(binomial(2*n+1,2*j)-2*n*binomial(n-1,j-1)))}

Formula

a(n) = Sum_{j=0..n} C(2*n-j-1, n-j)*( C(2*n+1, 2*j) - 2*n*C(n-1, j-1) ).

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