A238016 -id:A238016 - cp's OEIS frontend

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

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).

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).

A238634 Number of partitions of 7^n into parts that are at most 7.

Original entry on oeis.org

1, 15, 16475, 569704489, 54667189410224, 6242342067484101895, 731267824140098782358035, 85980297709044488588773397089, 10114611726199237476675435354424104, 1189959092808570377265545326042454670975, 139997247522791157386395916200494707946968395

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Row n=7 of A238016.

a(n) = [x^(7^n)] Product_{j=1..7} 1/(1-x^j).

A238635 Number of partitions of 8^n into parts that are at most 8.

Original entry on oeis.org

1, 22, 116263, 57733506640, 98149884074667116, 200386212932492140762672, 418829370245413954052212657987, 877979540384895591800176962368065072, 1841159754991692001851990839259642586671980, 3861166489120966379893685013624485791901912419888

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Row n=8 of A238016.

a(n) = [x^(8^n)] Product_{j=1..8} 1/(1-x^j).

A238636 Number of partitions of 9^n into parts that are at most 9.

Original entry on oeis.org

1, 30, 845105, 6944433285769, 241192889005578902877, 10133053906998476170548376403, 435014756168760380909523387186194290, 18720322073127387624828552135278902045913865, 805821524592736878546553406787954567208757510893718

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Row n=9 of A238016.

a(n) = [x^(9^n)] Product_{j=1..9} 1/(1-x^j).

A238637 Number of partitions of 10^n into parts that are at most 10.

Original entry on oeis.org

1, 42, 6292069, 968356321790171, 778400276435728381405745, 761287353202857218355451068558296, 759593815557626617904440619008375351308296, 759424638305250205001161810310150848799911916053194, 759407722344103064122231022019913967203947808354408941053194

Offset: 0

Alois P. Heinz, Mar 01 2014

nonn

Row n=10 of A238016.

a(n) = [x^(10^n)] Product_{j=1..10} 1/(1-x^j).

cp's OEIS Frontend

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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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.

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Examples

Crossrefs

Formula

A238634 Number of partitions of 7^n into parts that are at most 7.

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Author

Keywords

Crossrefs

Formula

A238635 Number of partitions of 8^n into parts that are at most 8.

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Author

Keywords

Crossrefs

Formula

A238636 Number of partitions of 9^n into parts that are at most 9.

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Author

Keywords

Crossrefs

Formula

A238637 Number of partitions of 10^n into parts that are at most 10.

Views

Author

Keywords

Crossrefs

Formula