A329076 Constant term in the expansion of ((Sum_{k=-n..n} x^k) * (Sum_{k=-n..n} y^k) - (Sum_{k=-n+1..n-1} x^k) * (Sum_{k=-n+1..n-1} y^k))^n.

1, 0, 16, 72, 7008, 162000, 17555520, 1093527120, 140846184640, 16016249944800, 2550757928818680, 419682645514181280, 82389928294166805312, 17418502084657134228768, 4123280170924828458697152, 1054943518137131171386437600, 293933660095874311773617934720, 87968971083026619734709639853632

Offset: 0

Author

Seiichi Manyama, Nov 04 2019

Keywords

• nonn
• walk

Comments

Also number of n-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 2*n).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..150 (terms 0..53 from Vaclav Kotesovec)
• Wikipedia, Taxicab geometry.

Crossrefs

Main diagonal of A329074.

Cf. A094061, A286928.

Programs

PARI

{a(n) = polcoef(polcoef((sum(k=-n, n, x^k)*sum(k=-n, n, y^k)-sum(k=-n+1, n-1, x^k)*sum(k=-n+1, n-1, y^k))^n, 0), 0)}

PARI

{a(n) = polcoef(polcoef((sum(k=0, 2*n, (x^k+1/x^k)*(y^(2*n-k)+1/y^(2*n-k)))-x^(2*n)-1/x^(2*n)-y^(2*n)-1/y^(2*n))^n, 0), 0)}

PARI

f(n) = (x^(n+1)-1/x^n)/(x-1);
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef(f(n)^k*f(n-1)^(n-k), 0)^2)

Formula

Conjecture: a(n) ~ 3 * 2^(3*n - 2) * n^(n-3) / Pi. - Vaclav Kotesovec, Nov 05 2019