A329078 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the number of k-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = n).

1, 1, 1, 1, 0, 1, 1, 4, 0, 1, 1, 0, 8, 0, 1, 1, 36, 24, 12, 0, 1, 1, 0, 216, 0, 16, 0, 1, 1, 400, 1200, 588, 48, 20, 0, 1, 1, 0, 8840, 0, 1200, 0, 24, 0, 1, 1, 4900, 58800, 49440, 10200, 2100, 72, 28, 0, 1, 1, 0, 423640, 0, 165760, 0, 3336, 0, 32, 0, 1

Offset: 0

Author

Seiichi Manyama, Nov 04 2019

Keywords

• nonn
• tabl
• walk

Comments

T(n,k) is the constant term in the expansion of (Sum_{j=0..n} (x^j + 1/x^j)*(y^(n-j) + 1/y^(n-j)) - x^n - 1/x^n - y^n - 1/y^n)^k for n > 0.

Examples

			Square array begins:
   1, 1,  1,  1,    1,     1, ...
   1, 0,  4,  0,   36,     0, ...
   1, 0,  8, 24,  216,  1200, ...
   1, 0, 12,  0,  588,     0, ...
   1, 0, 16, 48, 1200, 10200, ...
   1, 0, 20,  0, 2100,     0, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..25, flattened
• Wikipedia, Taxicab geometry.

Crossrefs

Cf. A329066, A329074.

Programs

PARI

{T(n,k) = if(n==0, 1, polcoef(polcoef((sum(j=0, n, (x^j+1/x^j)*(y^(n-j)+1/y^(n-j)))-x^n-1/x^n-y^n-1/y^n)^k, 0), 0))}