A329075 -id:A329075 - 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.

A329067 Constant term in the expansion of ((x^5 + x^3 + x + 1/x + 1/x^3 + 1/x^5)(y^5 + y^3 + y + 1/y + 1/y^3 + 1/y^5) - (x^3 + x + 1/x + 1/x^3)(y^3 + y + 1/y + 1/y^3))^(2*n).

Original entry on oeis.org

1, 20, 2100, 423440, 117234740, 36938855520, 12321942357648, 4240628338620960, 1489773976776270900, 531369088429408040240, 191788135117910898767200, 69889981814391283195249872, 25671987914195551303751107472, 9493180954173722971961114187200

Offset: 0

Seiichi Manyama, Nov 03 2019

Also 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| = 5).

Seiichi Manyama, Table of n, a(n) for n = 0..300 (terms 0..100 from Vaclav Kotesovec)
Wikipedia, Taxicab geometry.

Row n=2 of A329066.

Cf. A094061, A329075.

{a(n) = polcoef(polcoef(((x^5+x^3+x+1/x+1/x^3+1/x^5)*(y^5+y^3+y+1/y+1/y^3+1/y^5)-(x^3+x+1/x+1/x^3)*(y^3+y+1/y+1/y^3))^(2*n), 0), 0)}

{a(n) = polcoef(polcoef((sum(k=0, 5, (x^k+1/x^k)*(y^(5-k)+1/y^(5-k)))-x^5-1/x^5-y^5-1/y^5)^(2*n), 0), 0)}

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

Conjecture: a(n) ~ 400^n / (17*Pi*n). - Vaclav Kotesovec, Nov 04 2019

A329077 Constant term in the expansion of ((Sum_{k=-3..3} x^k) * (Sum_{k=-3..3} y^k) - (Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k))^n.

Original entry on oeis.org

1, 0, 24, 72, 3336, 34800, 912840, 15661520, 355423880, 7241240160, 160151370624, 3461028611040, 76789098028104, 1700195813892576, 38037857914721808, 853169553940415712, 19240825799184080520, 435267116844063531456, 9882232970998312871232

Offset: 0

Seiichi Manyama, Nov 04 2019

Also number of n-step closed paths (from origin to origin) in 2-dimensional lattice, using steps (t_1,t_2) (|t_1| + |t_2| = 6).

Seiichi Manyama, Table of n, a(n) for n = 0..500 (terms 0..230 from Vaclav Kotesovec)
Wikipedia, Taxicab geometry.

Row n=3 of A329074.

Cf. A002894, A094061, A329024, A329067, A329075.

{a(n) = polcoef(polcoef((sum(k=-3, 3, x^k)*sum(k=-3, 3, y^k)-sum(k=-2, 2, x^k)*sum(k=-2, 2, y^k))^n, 0), 0)}

{a(n) = polcoef(polcoef((sum(k=0, 6, (x^k+1/x^k)*(y^(6-k)+1/y^(6-k)))-x^6-1/x^6-y^6-1/y^6)^n, 0), 0)}

f(n) = (x^(n+1)-1/x^n)/(x-1);
a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n, k)*polcoef(f(3)^k*f(2)^(n-k), 0)^2)

Conjecture: a(n) ~ 6 * 24^n / (73*Pi*n). - Vaclav Kotesovec, Nov 04 2019

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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A329067 Constant term in the expansion of ((x^5 + x^3 + x + 1/x + 1/x^3 + 1/x^5)(y^5 + y^3 + y + 1/y + 1/y^3 + 1/y^5) - (x^3 + x + 1/x + 1/x^3)(y^3 + y + 1/y + 1/y^3))^(2*n).

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A329077 Constant term in the expansion of ((Sum_{k=-3..3} x^k) * (Sum_{k=-3..3} y^k) - (Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k))^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

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A329067 Constant term in the expansion of ((x^5 + x^3 + x + 1/x + 1/x^3 + 1/x^5)*(y^5 + y^3 + y + 1/y + 1/y^3 + 1/y^5) - (x^3 + x + 1/x + 1/x^3)*(y^3 + y + 1/y + 1/y^3))^(2*n).

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Programs

PARI

PARI

PARI

Formula

A329077 Constant term in the expansion of ((Sum_{k=-3..3} x^k) * (Sum_{k=-3..3} y^k) - (Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k))^n.

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Author

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Comments

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Programs

PARI

PARI

PARI

Formula

A329067 Constant term in the expansion of ((x^5 + x^3 + x + 1/x + 1/x^3 + 1/x^5)(y^5 + y^3 + y + 1/y + 1/y^3 + 1/y^5) - (x^3 + x + 1/x + 1/x^3)(y^3 + y + 1/y + 1/y^3))^(2*n).