A329819 -id:A329819 - cp's OEIS frontend

A329816 Triangular array, read by rows: T(n,k) = [(xy)^k] (-1 + (1 + x + 1/x)(1 + y + 1/y))^n for -n <= k <= n.

1, 1, 0, 1, 1, 2, 8, 2, 1, 1, 6, 27, 24, 27, 6, 1, 1, 12, 70, 132, 216, 132, 70, 12, 1, 1, 20, 155, 480, 1070, 1200, 1070, 480, 155, 20, 1, 1, 30, 306, 1370, 4035, 6900, 8840, 6900, 4035, 1370, 306, 30, 1, 1, 42, 553, 3332, 12621, 29750, 51065, 58800, 51065, 29750, 12621, 3332, 553, 42, 1

Offset: 0

Seiichi Manyama, Nov 21 2019

Also the coefficient of (x/y)^k in the expansion of (-1 + (1 + x + 1/x)*(1 + y + 1/y))^n for -n <= k <= n.

T(n,k) is the number of n step walks a chess king can take from (0,0) to (k,k). For example, for n=3 starting from (0,0) there is 1 walk to (3,3), 6 walks to (2,2), 27 walks to (1,1), 24 walks to (0,0), 27 walks to (-1,-1), 6 walks to (-2,-2) and 1 walk to (-3,-3). - Martin Clever, May 27 2023

			-1 + (1 + x + 1/x)*(1 + y + 1/y) = x*y + 1/(x*y) + x/y + y/x + x + 1/x + y + 1/y. So T(1,-1) = 1, T(1,0) = 0, T(1,1) = 1.
Triangle begins:
                          1;
                    1,    0,    1;
              1,    2,    8,    2,   1;
         1,   6,   27,   24,   27,   6,   1;
    1,  12,  70,  132,  216,  132,  70,  12,  1;
1, 20, 155, 480, 1070, 1200, 1070, 480, 155, 20, 1;

Seiichi Manyama, Rows n = 0..50, flattened

T(n,0) gives A094061.
Row sums give A288470.
Cf. A260492, A329819, A329820.

{T(n, k) = polcoef(polcoef((-1+(1+x+1/x)*(1+y+1/y))^n, k), k)}

T(n,k) = T(n,-k).

A329820 Triangular array, read by rows: T(n,k) = [(wxyz)^k] (-1 + (1 + w + 1/w)(1 + x + 1/x)(1 + y + 1/y)(1 + z + 1/z))^n for -n <= k <= n.

Original entry on oeis.org

1, 1, 0, 1, 1, 14, 80, 14, 1, 1, 78, 1251, 2160, 1251, 78, 1, 1, 252, 9682, 60444, 121200, 60444, 9682, 252, 1, 1, 620, 49355, 760800, 3785750, 6136800, 3785750, 760800, 49355, 620, 1, 1, 1290, 190746, 5950070, 60898395, 228400980, 356570960, 228400980, 60898395, 5950070, 190746, 1290, 1

Offset: 0

Seiichi Manyama, Nov 21 2019

			Triangle begins:
                          1;
                  1,      0,     1;
           1,    14,     80,    14,    1;
     1,   78,  1251,   2160,  1251,   78,   1;
1, 252, 9682, 60444, 121200, 60444, 9682, 252, 1;

T(n,0) gives A328875.

Cf. A260492, A329816, A329819.

{T(n, k) = polcoef(polcoef(polcoef(polcoef((-1+(1+w+1/w)*(1+x+1/x)*(1+y+1/y)*(1+z+1/z))^n, k), k), k), k)}

T(n,k) = T(n,-k).

cp's OEIS Frontend

A329816 Triangular array, read by rows: T(n,k) = [(xy)^k] (-1 + (1 + x + 1/x)(1 + y + 1/y))^n for -n <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A329820 Triangular array, read by rows: T(n,k) = [(wxyz)^k] (-1 + (1 + w + 1/w)(1 + x + 1/x)(1 + y + 1/y)(1 + z + 1/z))^n for -n <= k <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

cp's OEIS Frontend

A329816 Triangular array, read by rows: T(n,k) = [(x*y)^k] (-1 + (1 + x + 1/x)*(1 + y + 1/y))^n for -n <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A329820 Triangular array, read by rows: T(n,k) = [(w*x*y*z)^k] (-1 + (1 + w + 1/w)*(1 + x + 1/x)*(1 + y + 1/y)*(1 + z + 1/z))^n for -n <= k <= n.

Views

Author

Keywords

Examples

Crossrefs

Programs

PARI

Formula

A329816 Triangular array, read by rows: T(n,k) = [(xy)^k] (-1 + (1 + x + 1/x)(1 + y + 1/y))^n for -n <= k <= n.

A329820 Triangular array, read by rows: T(n,k) = [(wxyz)^k] (-1 + (1 + w + 1/w)(1 + x + 1/x)(1 + y + 1/y)(1 + z + 1/z))^n for -n <= k <= n.