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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
1, 12, 2100, 1751680, 4190017860, 20874801722544, 177661172742061008, 2295966445175463883680, 41848194615009705993547620, 1022849138778659709119846990032, 32304962696573489860535097887683296 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

Table of n, a(n) for n=0..10.

Wikipedia, Taxicab geometry.

PROG

(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);

CROSSREFS

Main diagonal of A329066.

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

Sequence in context: A208252 A204622 A004823 * A009063 A012675 A175014

Adjacent sequences:  A342961 A342962 A342963 * A342965 A342966 A342967

KEYWORD

nonn

AUTHOR

Seiichi Manyama, Mar 31 2021

STATUS

approved

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Last modified October 22 23:20 EDT 2021. Contains 348181 sequences. (Running on oeis4.)