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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.

%I A327751 #63 Oct 30 2019 11:13:35
%S A327751 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,
%T A327751 1200,5646,2160,242,0,1,0,0,8840,101520,121200,16080,728,0,1,0,70,
%U A327751 58800,2103740,6136800,2410326,115464,2186,0,1
%N 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.
%C A327751 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).
%H A327751 Seiichi Manyama, <a href="/A327751/b327751.txt">Antidiagonals n = 0..93, flattened</a>
%F A327751 T(n,k) = Sum_{j=0..k} (-1)^(k-j) * binomial(k,j) * A002426(j)^n.
%e A327751 Square array begins:
%e A327751    1, 0,   0,     0,       0,         0, ...
%e A327751    1, 0,   2,     0,       6,         0, ...
%e A327751    1, 0,   8,    24,     216,      1200, ...
%e A327751    1, 0,  26,   264,    5646,    101520, ...
%e A327751    1, 0,  80,  2160,  121200,   6136800, ...
%e A327751    1, 0, 242, 16080, 2410326, 332810400, ...
%Y A327751 Columns k=0-3 give A000012, A000004, A024023, 24*A016212(n-2).
%Y A327751 Rows n=0-4 give A000007, A126869, A094061, A328874, A328875.
%Y A327751 Main diagonal is A326920.
%Y A327751 Cf. A002426, A328718.
%K A327751 nonn,tabl
%O A327751 0,8
%A A327751 _Seiichi Manyama_, Oct 30 2019