A347606 -id:A347606 - cp's OEIS frontend

A347605 Number of partitions of n^n into n or more parts.

Original entry on oeis.org

1, 1, 4, 2996, 365749566865192, 8630901377559029573671524821295260243701883575513285157821

Offset: 0

Seiichi Manyama, Sep 08 2021

nonn

The next term is too large to include.

Cf. A238000, A347585, A347606, A347607.

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

a(n) = A347607(n) + A347606(n) - A238000(n).

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.

Original entry on oeis.org

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

Seiichi Manyama, Sep 08 2021

			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, ...

Columns k=0..3 give A063524, A000012, A206240, A304176.
Main diagonal gives A347606.
Cf. A238016, A347615, A347618.

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));

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

cp's OEIS Frontend

A347605 Number of partitions of n^n into n or more parts.

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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.

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