A347617 Square array T(n,k), n >= 0, k >= 0, read by antidiagonals downwards, where T(n,k) is the number of partitions of n^k into exactly n parts.

0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 4, 7, 1, 0, 1, 1, 8, 61, 34, 1, 0, 1, 1, 16, 547, 1906, 192, 1, 0, 1, 1, 32, 4921, 117874, 91606, 1206, 1, 0, 1, 1, 64, 44287, 7478386, 53830967, 6023602, 8033, 1, 0, 1, 1, 128, 398581, 477568114, 33219689231, 43054503928, 505853354, 55974, 1, 0

Offset

0,13

Links

Table of n, a(n) for n=0..65.

Formula

T(n,k) = [x^(n^k-n)] Product_{j=1..n} 1/(1-x^j).

Example

Square array begins:
  0, 1,   1,     1,        1,           1, ...
  1, 1,   1,     1,        1,           1, ...
  0, 1,   2,     4,        8,          16, ...
  0, 1,   7,    61,      547,        4921, ...
  0, 1,  34,  1906,   117874,     7478386, ...
  0, 1, 192, 91606, 53830967, 33219689231, ...

Prog

(PARI) T(n, k) = if(k==0, n==1, polcoef(prod(j=1, n, 1/(1-x^j+x*O(x^(n^k-n)))), n^k-n));

Crossrefs

Columns k=0..3 give A063524, A000012, A206240, A304176.

Main diagonal gives A347606.

Cf. A238016, A347615, A347618.

Sequence in context: A355262 A355576 A198062 * A226690 A318557 A245683

Adjacent sequences: A347614 A347615 A347616 * A347618 A347619 A347620

Keyword

nonn,tabl

Author

Seiichi Manyama, Sep 08 2021

Status

approved