A329816 - OEIS

%I #40 Jul 04 2023 10:01:08

%S 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,

%T 155,480,1070,1200,1070,480,155,20,1,1,30,306,1370,4035,6900,8840,

%U 6900,4035,1370,306,30,1,1,42,553,3332,12621,29750,51065,58800,51065,29750,12621,3332,553,42,1

%N 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.

%C 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.

%C 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

%H Seiichi Manyama, <a href="/A329816/b329816.txt">Rows n = 0..50, flattened</a>

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

%e -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.

%e Triangle begins:

%e                           1;

%e                     1,    0,    1;

%e               1,    2,    8,    2,   1;

%e          1,   6,   27,   24,   27,   6,   1;

%e     1,  12,  70,  132,  216,  132,  70,  12,  1;

%e 1, 20, 155, 480, 1070, 1200, 1070, 480, 155, 20, 1;

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

%Y T(n,0) gives A094061.

%Y Row sums give A288470.

%Y Cf. A260492, A329819, A329820.

%K nonn,tabf

%O 0,6

%A _Seiichi Manyama_, Nov 21 2019