A329816 -id:A329816 - cp's OEIS frontend

A329819 Triangular array, read by rows: T(n,k) = [(xyz)^k] (-1 + (1 + x + 1/x)(1 + y + 1/y)(1 + z + 1/z))^n for -n <= k <= n.

1, 1, 0, 1, 1, 6, 26, 6, 1, 1, 24, 195, 264, 195, 24, 1, 1, 60, 898, 3276, 5646, 3276, 898, 60, 1, 1, 120, 3065, 22260, 72730, 101520, 72730, 22260, 3065, 120, 1, 1, 210, 8526, 105690, 581475, 1510860, 2103740, 1510860, 581475, 105690, 8526, 210, 1

Offset: 0

Seiichi Manyama, Nov 21 2019

			Triangle begins:
                                 1;
                         1,      0,     1;
                  1,     6,     26,     6,     1;
           1,    24,   195,    264,   195,    24,    1;
     1,   60,   898,  3276,   5646,  3276,   898,   60,   1;
1, 120, 3065, 22260, 72730, 101520, 72730, 22260, 3065, 120, 1;

Seiichi Manyama, Rows n = 0..25, flattened

T(n,0) gives A328874.

Cf. A260492, A329816, A329820.

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

A329819 Triangular array, read by rows: T(n,k) = [(xyz)^k] (-1 + (1 + x + 1/x)(1 + y + 1/y)(1 + z + 1/z))^n for -n <= k <= n.

Views

Author

Keywords

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

A329819 Triangular array, read by rows: T(n,k) = [(x*y*z)^k] (-1 + (1 + x + 1/x)*(1 + y + 1/y)*(1 + z + 1/z))^n for -n <= k <= n.

Views

Author

Keywords

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

A329819 Triangular array, read by rows: T(n,k) = [(xyz)^k] (-1 + (1 + x + 1/x)(1 + y + 1/y)(1 + z + 1/z))^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.