A329067 -id:A329067 - 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

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

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

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

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.

Views

Author

Keywords

Comments

Links

Crossrefs

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.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

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).

Views

Author

Keywords

Comments

Links

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).

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).