A329074 - OEIS

A329074

Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ((Sum_{j=-n..n} x^j) * (Sum_{j=-n..n} y^j) - (Sum_{j=-n+1..n-1} x^j) * (Sum_{j=-n+1..n-1} y^j))^k.

1, 1, 1, 1, 0, 1, 1, 8, 0, 1, 1, 24, 16, 0, 1, 1, 216, 48, 24, 0, 1, 1, 1200, 1200, 72, 32, 0, 1, 1, 8840, 10200, 3336, 96, 40, 0, 1, 1, 58800, 165760, 34800, 7008, 120, 48, 0, 1, 1, 423640, 2032800, 912840, 82800, 12600, 144, 56, 0, 1

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OFFSET

0,8

COMMENTS

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| = 2*n).

T(n,k) is the constant term in the expansion of (Sum_{j=0..2*n} (x^j + 1/x^j)*(y^(2*n-j) + 1/y^(2*n-j)) - x^(2*n) - 1/x^(2*n) - y^(2*n) - 1/y^(2*n))^k for n > 0.

LINKS

Seiichi Manyama, Antidiagonals n = 0..100, flattened

Wikipedia, Taxicab geometry.

FORMULA

T(0,k) = 1^k = 1.

See the second code written in PARI.

EXAMPLE

Square array begins:

1, 1, 1, 1, 1, 1, ...

1, 0, 8, 24, 216, 1200, ...

1, 0, 16, 48, 1200, 10200, ...

1, 0, 24, 72, 3336, 34800, ...