A342964 Constant term in the expansion of ( (Sum_{j=0..n} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n} y^(2*j+1)+1/y^(2*j+1)) - (Sum_{j=0..n-1} x^(2*j+1)+1/x^(2*j+1)) * (Sum_{j=0..n-1} y^(2*j+1)+1/y^(2*j+1)) )^(2*n).

1, 12, 2100, 1751680, 4190017860, 20874801722544, 177661172742061008, 2295966445175463883680, 41848194615009705993547620, 1022849138778659709119846990032, 32304962696573489860535097887683296

Offset: 0

Author

Seiichi Manyama, Mar 31 2021

Keywords

• nonn

Comments

Number of (2*n)-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 2*n+1).

Constant term in the expansion of (Sum_{j=0..2*n+1} (x^j + 1/x^j)*(y^(2*n+1-j) + 1/y^(2*n+1-j)) - x^(2*n+1) - 1/x^(2*n+1) - y^(2*n+1) - 1/y^(2*n+1))^(2*n).

Links

• Wikipedia, Taxicab geometry.

Crossrefs

Main diagonal of A329066.

Cf. A002894, A328716, A329024, A329067, A329076.

Programs

PARI

f(n) = (x^(2*n+2)-1/x^(2*n+2))/(x-1/x);
a(n) = sum(j=0, 2*n, (-1)^j*binomial(2*n, j)*polcoef(f(n)^j*f(n-1)^(2*n-j), 0)^2);