cp's OEIS Frontend

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).

%I A329078 #21 Nov 04 2019 07:25:18
%S A329078 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,
%T A329078 1200,588,48,20,0,1,1,0,8840,0,1200,0,24,0,1,1,4900,58800,49440,10200,
%U A329078 2100,72,28,0,1,1,0,423640,0,165760,0,3336,0,32,0,1
%N 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).
%C A329078 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.
%H A329078 Seiichi Manyama, <a href="/A329078/b329078.txt">Antidiagonals n = 0..25, flattened</a>
%H A329078 Wikipedia, <a href="https://en.wikipedia.org/wiki/Taxicab_geometry">Taxicab geometry</a>.
%e A329078 Square array begins:
%e A329078    1, 1,  1,  1,    1,     1, ...
%e A329078    1, 0,  4,  0,   36,     0, ...
%e A329078    1, 0,  8, 24,  216,  1200, ...
%e A329078    1, 0, 12,  0,  588,     0, ...
%e A329078    1, 0, 16, 48, 1200, 10200, ...
%e A329078    1, 0, 20,  0, 2100,     0, ...
%o A329078 (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))}
%Y A329078 Cf. A329066, A329074.
%K A329078 nonn,tabl,walk
%O A329078 0,8
%A A329078 _Seiichi Manyama_, Nov 04 2019