A329067 Constant term in the expansion of ((x^5 + x^3 + x + 1/x + 1/x^3 + 1/x^5)*(y^5 + y^3 + y + 1/y + 1/y^3 + 1/y^5) - (x^3 + x + 1/x + 1/x^3)*(y^3 + y + 1/y + 1/y^3))^(2*n).

1, 20, 2100, 423440, 117234740, 36938855520, 12321942357648, 4240628338620960, 1489773976776270900, 531369088429408040240, 191788135117910898767200, 69889981814391283195249872, 25671987914195551303751107472, 9493180954173722971961114187200

Offset: 0

Author

Seiichi Manyama, Nov 03 2019

Keywords

• nonn
• walk

Comments

Also number of (2*n)-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 5).

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..300 (terms 0..100 from Vaclav Kotesovec)
• Wikipedia, Taxicab geometry.

Crossrefs

Row n=2 of A329066.

Cf. A094061, A329075.

Programs

PARI

{a(n) = polcoef(polcoef(((x^5+x^3+x+1/x+1/x^3+1/x^5)*(y^5+y^3+y+1/y+1/y^3+1/y^5)-(x^3+x+1/x+1/x^3)*(y^3+y+1/y+1/y^3))^(2*n), 0), 0)}

PARI

{a(n) = polcoef(polcoef((sum(k=0, 5, (x^k+1/x^k)*(y^(5-k)+1/y^(5-k)))-x^5-1/x^5-y^5-1/y^5)^(2*n), 0), 0)}

PARI

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

Formula

Conjecture: a(n) ~ 400^n / (17*Pi*n). - Vaclav Kotesovec, Nov 04 2019