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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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.
Sequence in context: A011309 A087198 A200589 * A194567 A351794 A065813
KEYWORD
nonn,tabf
AUTHOR
Seiichi Manyama, Nov 21 2019
STATUS
approved

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Last modified March 29 05:43 EDT 2024. Contains 371264 sequences. (Running on oeis4.)