A245243 Triangle, read by rows, defined by T(n,k) = C(n^2 - k^2, n*k - k^2), for k=0..n, n>=0.

1, 1, 1, 1, 3, 1, 1, 28, 10, 1, 1, 455, 495, 35, 1, 1, 10626, 54264, 8008, 126, 1, 1, 324632, 10518300, 4686825, 125970, 462, 1, 1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1, 1, 553270671, 1399358844975, 11899700525790, 2254848913647, 25140840660, 30421755, 6435, 1

Offset

0,5

Comments

Row sums equal A245242.

Central terms are A245245(n) = C(3*n^2, n^2).

Links

Paul D. Hanna, Table of n, a(n) for rows 0..30 of flattened triangle.

Formula

T(n,k) = C(n^2, n*k) * C(n*k, k^2) / C(n^2, k^2).

T(n,k) = (n^2 - k^2)! / ( (n^2 - n*k)! * (n*k - k^2)! ).

T(n,k) = ((n+k)*(n-k))! / ( (n*(n-k))! * (k*(n-k))! ).

Example

Triangle T(n,k) = C(n^2 - k^2, n*k - k^2) begins:
1;
1, 1;
1, 3, 1;
1, 28, 10, 1;
1, 455, 495, 35, 1;
1, 10626, 54264, 8008, 126, 1;
1, 324632, 10518300, 4686825, 125970, 462, 1;
1, 12271512, 3190187286, 5586853480, 354817320, 1961256, 1716, 1;
1, 553270671, 1399358844975, 11899700525790, 2254848913647, 25140840660, 30421755, 6435, 1; ...

Mathematica

Table[Binomial[n^2-k^2, n k-k^2], {n, 0, 10}, {k, 0, n}]//Flatten (* Harvey P. Dale, Jan 06 2019 *)

Prog

(PARI) {T(n, k) = binomial(n^2 - k^2, n*k - k^2)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) {T(n, k) = binomial(n^2, n*k) * binomial(n*k, k^2) / binomial(n^2, k^2)}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))

Crossrefs

Cf. A245242 (row sums), A245245 (central terms), A209330, A228832.

Sequence in context: A083998 A277170 A216922 * A168242 A332540 A215750

Adjacent sequences: A245240 A245241 A245242 * A245244 A245245 A245246

Keyword

nonn,tabl

Author

Paul D. Hanna, Jul 14 2014

Status

approved