A329067 - OEIS

%I #33 Nov 05 2019 14:07:47

%S 1,20,2100,423440,117234740,36938855520,12321942357648,

%T 4240628338620960,1489773976776270900,531369088429408040240,

%U 191788135117910898767200,69889981814391283195249872,25671987914195551303751107472,9493180954173722971961114187200

%N 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).

%C 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).

%H Seiichi Manyama, <a href="/A329067/b329067.txt">Table of n, a(n) for n = 0..300</a> (terms 0..100 from Vaclav Kotesovec)

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Taxicab_geometry">Taxicab geometry</a>.

%F Conjecture: a(n) ~ 400^n / (17*Pi*n). - _Vaclav Kotesovec_, Nov 04 2019

%o (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)}

%o (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)}

%o (PARI) f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);

%o a(n) = sum(k=0, 2*n, (-1)^k*binomial(2*n, k)*polcoef(f(2)^k*f(1)^(2*n-k), 0)^2)

%Y Row n=2 of A329066.

%Y Cf. A094061, A329075.

%K nonn,walk

%O 0,2

%A _Seiichi Manyama_, Nov 03 2019