A329066 -id:A329066 - 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.

A329024 Constant term in the expansion of ((x^3 + x + 1/x + 1/x^3)(y^3 + y + 1/y + 1/y^3) - (x + 1/x)(y + 1/y))^(2*n).

Original entry on oeis.org

1, 12, 588, 49440, 5187980, 597027312, 71962945824, 8923789535232, 1128795397492620, 144940851928720848, 18832163401980525168, 2470451402766989534256, 326667449725835512275488, 43485599433527022301377600, 5821983056232777427055717760

Offset: 0

Seiichi Manyama, Nov 02 2019

nonn

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| = 3).
         *
         |
      *-- --*
      |  |  |
   *-- -- -- --*
   |  |  |  |  |
*-- -- --P-- -- --*
   |  |  |  |  |
   *-- -- -- --*
      |  |  |
      *-- --*
         |
         *
Point P move to any position of * in the next step.

Seiichi Manyama, Table of n, a(n) for n = 0..400 (terms 0..185 from Vaclav Kotesovec)
Vaclav Kotesovec, Recurrence of order 4 (conjectured)

Row n=1 of A329066.

Cf. A002894, A094061, A254129.

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

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

Conjecture: a(n) ~ 3 * 144^n / (19*Pi*n). - Vaclav Kotesovec, Nov 04 2019

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

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))}

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

PARI

Formula

A329024 Constant term in the expansion of ((x^3 + x + 1/x + 1/x^3)*(y^3 + y + 1/y + 1/y^3) - (x + 1/x)*(y + 1/y))^(2*n).

Views

Author

Keywords

Comments

Links

Crossrefs

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).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

PARI

PARI

Formula

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).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

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

A329024 Constant term in the expansion of ((x^3 + x + 1/x + 1/x^3)(y^3 + y + 1/y + 1/y^3) - (x + 1/x)(y + 1/y))^(2*n).

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).

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).