A329076 -id:A329076 - cp's OEIS frontend

A329074 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ((Sum_{j=-n..n} x^j) * (Sum_{j=-n..n} y^j) - (Sum_{j=-n+1..n-1} x^j) * (Sum_{j=-n+1..n-1} y^j))^k.

1, 1, 1, 1, 0, 1, 1, 8, 0, 1, 1, 24, 16, 0, 1, 1, 216, 48, 24, 0, 1, 1, 1200, 1200, 72, 32, 0, 1, 1, 8840, 10200, 3336, 96, 40, 0, 1, 1, 58800, 165760, 34800, 7008, 120, 48, 0, 1, 1, 423640, 2032800, 912840, 82800, 12600, 144, 56, 0, 1

Offset: 0

Seiichi Manyama, Nov 03 2019

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| = 2*n).

T(n,k) is the constant term in the expansion of (Sum_{j=0..2*n} (x^j + 1/x^j)*(y^(2*n-j) + 1/y^(2*n-j)) - x^(2*n) - 1/x^(2*n) - y^(2*n) - 1/y^(2*n))^k for n > 0.

			Square array begins:
   1, 1,  1,   1,     1,      1, ...
   1, 0,  8,  24,   216,   1200, ...
   1, 0, 16,  48,  1200,  10200, ...
   1, 0, 24,  72,  3336,  34800, ...
   1, 0, 32,  96,  7008,  82800, ...
   1, 0, 40, 120, 12600, 162000, ...

Seiichi Manyama, Antidiagonals n = 0..100, flattened
Wikipedia, Taxicab geometry.

Rows n=0-3 give A000012, A094061, A329075, A329077.
Main diagonal gives A329076.
Cf. A329066.

{T(n, k) = if(n==0, 1, polcoef(polcoef((sum(j=0, 2*n, (x^j+1/x^j)*(y^(2*n-j)+1/y^(2*n-j)))-x^(2*n)-1/x^(2*n)-y^(2*n)-1/y^(2*n))^k, 0), 0))}

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

T(0,k) = 1^k = 1.

See the second code written in PARI.

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

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Mar 31 2021

nonn

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).

Wikipedia, Taxicab geometry.

Main diagonal of A329066.

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

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);

cp's OEIS Frontend

A329074 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ((Sum_{j=-n..n} x^j) * (Sum_{j=-n..n} y^j) - (Sum_{j=-n+1..n-1} x^j) * (Sum_{j=-n+1..n-1} y^j))^k.

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

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cp's OEIS Frontend

A329074 Square array T(n,k), n>=0, k>=0, read by antidiagonals, where T(n,k) is the constant term in the expansion of ((Sum_{j=-n..n} x^j) * (Sum_{j=-n..n} y^j) - (Sum_{j=-n+1..n-1} x^j) * (Sum_{j=-n+1..n-1} y^j))^k.

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PARI

PARI

Formula

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).

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PARI

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