A327751 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of (-1 + Product_{j=1..n} (1 + x_j + 1/x_j))^k.

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 8, 0, 1, 0, 6, 24, 26, 0, 1, 0, 0, 216, 264, 80, 0, 1, 0, 20, 1200, 5646, 2160, 242, 0, 1, 0, 0, 8840, 101520, 121200, 16080, 728, 0, 1, 0, 70, 58800, 2103740, 6136800, 2410326, 115464, 2186, 0, 1

Offset: 0

Author

Seiichi Manyama, Oct 30 2019

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of k-step closed walks (from origin to origin) in n-dimensional lattice, using steps (t_1,t_2, ... ,t_n) (t_j = -1, 1 or 0 for 1 <= j <= n) except for (0,0, ... ,0) (t_j = 0 for 1 <= j <= n).

Examples

			Square array begins:
   1, 0,   0,     0,       0,         0, ...
   1, 0,   2,     0,       6,         0, ...
   1, 0,   8,    24,     216,      1200, ...
   1, 0,  26,   264,    5646,    101520, ...
   1, 0,  80,  2160,  121200,   6136800, ...
   1, 0, 242, 16080, 2410326, 332810400, ...
		

Links

• Seiichi Manyama, Antidiagonals n = 0..93, flattened

Crossrefs

Columns k=0-3 give A000012, A000004, A024023, 24*A016212(n-2).

Rows n=0-4 give A000007, A126869, A094061, A328874, A328875.

Main diagonal is A326920.

Cf. A002426, A328718.

Formula

T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * A002426(j)^n.