A347617 -id:A347617 - cp's OEIS frontend

A347615 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.

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 5, 3, 1, 1, 1, 22, 30, 5, 1, 1, 1, 231, 3010, 231, 7, 1, 1, 1, 8349, 18004327, 1741630, 1958, 11, 1, 1, 1, 1741630, 133978259344888, 365749566870782, 3163127352, 17977, 15, 1, 1, 1, 4351078600, 233202632378520643600875145, 61847822068260244309086870983975, 1606903190858354689128371, 15285151248481, 173525, 22, 1

Offset: 0

Seiichi Manyama, Sep 08 2021

			Square array begins:
  1, 1,   1,       1,               1, ...
  1, 1,   1,       1,               1, ...
  1, 2,   5,      22,             231, ...
  1, 3,  30,    3010,        18004327, ...
  1, 5, 231, 1741630, 365749566870782, ...

Columns k=0..3 give A000012, A000041, A072213, A128854.
Rows n=0+1, 2-10 give A000012, A068413, A248728, A068413(2*n), A248730, A248732, A248734, A068413(3*n), A248728(2*n), A070177.
Main diagonal gives A347607.
Cf. A238016, A347617, A347618, A347621.

PARI
```
T(n, k) = numbpart(n^k);
```

T(n,k) = A000041(n^k).

A347618 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 n or more parts.

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 4, 1, 0, 1, 1, 21, 25, 1, 0, 1, 1, 230, 2996, 201, 1, 0, 1, 1, 8348, 18004286, 1741256, 1773, 1, 0, 1, 1, 1741629, 133978259344766, 365749566865192, 3163112106, 16751, 1, 0, 1, 1, 4351078599, 233202632378520643600874780, 61847822068260244309086870896081, 1606903190858354687391986, 15285150382556, 165083, 1, 0

Offset: 0

Seiichi Manyama, Sep 08 2021

			Square array begins:
  1, 1,   1,       1,               1, ...
  1, 1,   1,       1,               1, ...
  0, 1,   4,      21,             230, ...
  0, 1,  25,    2996,        18004286, ...
  0, 1, 201, 1741256, 365749566865192, ...

Columns k=0..3 give A019590(n+1), A000012, A347585, A347604.
Main diagonal gives A347605.
Cf. A238016, A347615, A347617.

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

T(n,k) = A347615(n,k) + A347617(n,k) - A238016(n,k).

cp's OEIS Frontend

A347615 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.

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