A329075 Constant term in the expansion of ((Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k) - (Sum_{k=-1..1} x^k) * (Sum_{k=-1..1} y^k))^n.

1, 0, 16, 48, 1200, 10200, 165760, 2032800, 30115120, 417189360, 6116225976, 88579001280, 1308168101856, 19335388664592, 288264711738432, 4311842765438208, 64819095869951280, 977630677389002208, 14796595755047824432, 224583060859608559680, 3417918348978709970680

Offset: 0

Author

Seiichi Manyama, Nov 03 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| = 4).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..500 (terms 0..300 from Vaclav Kotesovec)
• Vaclav Kotesovec, Recurrence of order 8 (conjectured)
• Wikipedia, Taxicab geometry.

Crossrefs

Row n=2 of A329074.

Cf. A094061, A329067.

Programs

PARI

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

PARI

{a(n) = polcoef(polcoef((sum(k=0, 4, (x^k+1/x^k)*(y^(4-k)+1/y^(4-k)))-x^4-1/x^4-y^4-1/y^4)^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(2)^k*f(1)^(n-k), 0)^2)

Formula

Conjecture: a(n) ~ 2 * 16^n / (11*Pi*n). - Vaclav Kotesovec, Nov 04 2019