A209330 Triangle defined by T(n,k) = binomial(n^2, n*k), for n>=0, k=0..n, as read by rows.

1, 1, 1, 1, 6, 1, 1, 84, 84, 1, 1, 1820, 12870, 1820, 1, 1, 53130, 3268760, 3268760, 53130, 1, 1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1, 1, 85900584, 675248872536, 39049918716424, 39049918716424, 675248872536, 85900584, 1, 1

Offset

0,5

Comments

Column 1 equals A014062.

Row sums equal A167009.

Antidiagonal sums equal A209331.

Ignoring initial row T(0,0), equals the logarithmic derivative of the g.f. of triangle A209196.

Links

Paul D. Hanna, Rows n = 0..30, as a flattened table of n, a(n) for n = 0..495

Example

The triangle of coefficients C(n^2,n*k), n>=k, k=0..n, begins:
1;
1, 1;
1, 6, 1;
1, 84, 84, 1;
1, 1820, 12870, 1820, 1;
1, 53130, 3268760, 3268760, 53130, 1;
1, 1947792, 1251677700, 9075135300, 1251677700, 1947792, 1;
1, 85900584, 675248872536, 39049918716424, 39049918716424, 675248872536, 85900584, 1; ...

Mathematica

Table[Binomial[n^2, n*k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jan 05 2018 *)

Prog

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

Crossrefs

Cf. A014062 (column 1), A167009 (row sums), A209331, A209196.

Cf. related triangles: A209196 (exp), A228836, A228832, A226234.

Cf. A206830.

Sequence in context: A172375 A075377 A046792 * A357297 A111825 A085552

Adjacent sequences: A209327 A209328 A209329 * A209331 A209332 A209333

Keyword

nonn,tabl

Author

Paul D. Hanna, Mar 06 2012

Status

approved