A329074 -id:A329074 - cp's OEIS frontend

A329066 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=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*k).

1, 4, 1, 36, 12, 1, 400, 588, 20, 1, 4900, 49440, 2100, 28, 1, 63504, 5187980, 423440, 4956, 36, 1, 853776, 597027312, 117234740, 1751680, 9540, 44, 1, 11778624, 71962945824, 36938855520, 907687900, 5101200, 16236, 52, 1

Offset: 0

Seiichi Manyama, Nov 03 2019

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

T(n,k) is the 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*k).

			Square array begins:
   1,  4,   36,     400,       4900, ...
   1, 12,  588,   49440,    5187980, ...
   1, 20, 2100,  423440,  117234740, ...
   1, 28, 4956, 1751680,  907687900, ...
   1, 36, 9540, 5101200, 4190017860, ...

Seiichi Manyama, Antidiagonals n = 0..50, flattened
Wikipedia, Taxicab geometry.

Columns k=0-1 give A000012, A017113.
Rows n=0-2 give A002894, A329024, A329067.
Main diagonal gives A342964.
Cf. A005408, A328718, A329074, A329078.

{T(n, k) = polcoef(polcoef((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*k), 0), 0)}

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

See the second code written in PARI.

A329075 Constant term in the expansion of ((Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k) - (Sum_{k=-1..1} x^k) * (Sum_{k=-1..1} y^k))^n.

Original entry on oeis.org

1, 0, 16, 48, 1200, 10200, 165760, 2032800, 30115120, 417189360, 6116225976, 88579001280, 1308168101856, 19335388664592, 288264711738432, 4311842765438208, 64819095869951280, 977630677389002208, 14796595755047824432, 224583060859608559680, 3417918348978709970680

Offset: 0

Seiichi Manyama, Nov 03 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| = 4).

Seiichi Manyama, Table of n, a(n) for n = 0..500 (terms 0..300 from Vaclav Kotesovec)
Vaclav Kotesovec, Recurrence of order 8 (conjectured)
Wikipedia, Taxicab geometry.

Row n=2 of A329074.

Cf. A094061, A329067.

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

{a(n) = polcoef(polcoef((sum(k=0, 4, (x^k+1/x^k)*(y^(4-k)+1/y^(4-k)))-x^4-1/x^4-y^4-1/y^4)^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(2)^k*f(1)^(n-k), 0)^2)

Conjecture: a(n) ~ 2 * 16^n / (11*Pi*n). - Vaclav Kotesovec, Nov 04 2019

A329076 Constant term in the expansion of ((Sum_{k=-n..n} x^k) * (Sum_{k=-n..n} y^k) - (Sum_{k=-n+1..n-1} x^k) * (Sum_{k=-n+1..n-1} y^k))^n.

Original entry on oeis.org

1, 0, 16, 72, 7008, 162000, 17555520, 1093527120, 140846184640, 16016249944800, 2550757928818680, 419682645514181280, 82389928294166805312, 17418502084657134228768, 4123280170924828458697152, 1054943518137131171386437600, 293933660095874311773617934720, 87968971083026619734709639853632

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| = 2*n).

Seiichi Manyama, Table of n, a(n) for n = 0..150 (terms 0..53 from Vaclav Kotesovec)
Wikipedia, Taxicab geometry.

Main diagonal of A329074.

Cf. A094061, A286928.

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

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

Conjecture: a(n) ~ 3 * 2^(3*n - 2) * n^(n-3) / Pi. - Vaclav Kotesovec, Nov 05 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

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).

Original entry on oeis.org

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, 1200, 588, 48, 20, 0, 1, 1, 0, 8840, 0, 1200, 0, 24, 0, 1, 1, 4900, 58800, 49440, 10200, 2100, 72, 28, 0, 1, 1, 0, 423640, 0, 165760, 0, 3336, 0, 32, 0, 1

Offset: 0

Seiichi Manyama, Nov 04 2019

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.

			Square array begins:
   1, 1,  1,  1,    1,     1, ...
   1, 0,  4,  0,   36,     0, ...
   1, 0,  8, 24,  216,  1200, ...
   1, 0, 12,  0,  588,     0, ...
   1, 0, 16, 48, 1200, 10200, ...
   1, 0, 20,  0, 2100,     0, ...

Seiichi Manyama, Antidiagonals n = 0..25, flattened
Wikipedia, Taxicab geometry.

Cf. A329066, A329074.

{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))}

cp's OEIS Frontend

A329066 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=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*k).

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A329075 Constant term in the expansion of ((Sum_{k=-2..2} x^k) * (Sum_{k=-2..2} y^k) - (Sum_{k=-1..1} x^k) * (Sum_{k=-1..1} y^k))^n.

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A329076 Constant term in the expansion of ((Sum_{k=-n..n} x^k) * (Sum_{k=-n..n} y^k) - (Sum_{k=-n+1..n-1} x^k) * (Sum_{k=-n+1..n-1} y^k))^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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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).

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Examples

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Crossrefs

Programs

PARI

A329066 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=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*k).