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.

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

0,6

Comments

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

Links

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

Formula

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

Example

-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;

Prog

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

Crossrefs

T(n,0) gives A094061.

Row sums give A288470.

Cf. A260492, A329819, A329820.

Sequence in context: A011309 A087198 A200589 * A194567 A351794 A065813

Adjacent sequences: A329813 A329814 A329815 * A329817 A329818 A329819

Keyword

nonn,tabf

Author

Seiichi Manyama, Nov 21 2019

Status

approved