A238001 -id:A238001 - cp's OEIS frontend

A238000 Number of partitions of n^n into parts that are at most n.

0, 1, 3, 75, 123464, 33432635477, 2561606354507677872, 85980297709044488588773397089, 1841159754991692001851990839259642586671980, 34687845413783594101366282545316028561007822069601179170488

Offset: 0

Alois P. Heinz, Feb 16 2014

nonn

			a(1) = 1: 1.
a(2) = 3: 22, 211, 1111.
a(3) = 75: 333333333, ..., 111111111111111111111111111.

Alois P. Heinz, Table of n, a(n) for n = 0..27
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz) (arXiv:1108.4391 [math.CO])

Main diagonal of A238010 and A238016.

Cf. A236810, A237512, A237998, A237999, A238001.

a[n_] := SeriesCoefficient[Product[1/(1 - x^j), {j, 1, n}], {x, 0, n^n}];
a[0] = 0;
Table[a[n], {n, 0, 5}] (* Jean-François Alcover, Nov 03 2018 *)

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

a(n) ~ exp(2*n) * n^(n*(n-3)) / (2*Pi). - Vaclav Kotesovec, May 25 2015

A238012 Number A(n,k) of partitions of k^n into parts that are at most n with at least one part of each size; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 4, 2, 0, 0, 0, 1, 7, 48, 9, 0, 0, 0, 1, 12, 310, 3042, 119, 0, 0, 0, 1, 17, 1240, 109809, 1067474, 4935, 0, 0, 0, 1, 24, 3781, 1655004, 370702459, 2215932130, 596763, 0, 0, 0, 1, 31, 9633, 14942231, 32796849930, 13173778523786, 29012104252380, 211517867, 0, 0

Offset: 0

Alois P. Heinz, Feb 16 2014

In general, column k>=2 is asymptotic to k^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

			Square array A(n,k) begins:
  0, 0,   0,       0,         0,           0, ...
  0, 1,   1,       1,         1,           1, ...
  0, 0,   1,       4,         7,          12, ...
  0, 0,   2,      48,       310,        1240, ...
  0, 0,   9,    3042,    109809,     1655004, ...
  0, 0, 119, 1067474, 370702459, 32796849930, ...

Alois P. Heinz, Antidiagonals n = 0..43, flattened
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz), arXiv:1108.4391 [math.CO], 2011.

Columns k=0-10 give: A000004, A063524, A237999, A239162, A239163, A239164, A239165, A239166, A239167, A239168, A239169.
Rows n=0-2 give: A000004, A057427, A074148(k-1) for k>1.
Main diagonal gives A238001.
Cf. A238010.

A[0, 0] = 0;
A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, k^n - n(n+1)/2}];
Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Aug 18 2018, after Alois P. Heinz *)

A(n,k) = [x^(k^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

A237998 Number of partitions of 2^n into parts that are at most n.

Original entry on oeis.org

0, 1, 3, 10, 64, 831, 26207, 2239706, 567852809, 454241403975, 1192075219982204, 10510218491798860052, 315981966712495811700951, 32726459268483342710907384794, 11771239570056489326716955796095261, 14808470136486015545654676685321653888199

Offset: 0

Alois P. Heinz, Feb 16 2014

nonn

			a(1) = 1: 11.
a(2) = 3: 22, 211, 1111.
a(3) = 10: 332, 2222, 3221, 3311, 22211, 32111, 221111, 311111, 2111111, 11111111.

Alois P. Heinz, Table of n, a(n) for n = 0..62
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz) (arXiv:1108.4391 [math.CO])

Column k=2 of A238010.

Cf. A236810, A237512, A237999, A238000, A238001, A258672.

a[n_] := SeriesCoefficient[Product[1/(1 - x^j), {j, 1, n}], {x, 0, 2^n}];
Table[a[n], {n, 0, 12}] (* Jean-François Alcover, Nov 03 2018 *)

a(n) = [x^(2^n)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 2^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A237999 Number of partitions of 2^n into parts that are at most n with at least one part of each size.

Original entry on oeis.org

0, 1, 1, 2, 9, 119, 4935, 596763, 211517867, 224663223092, 734961197081208, 7614278809664610952, 256261752606028225485183, 28642174350851846128820426827, 10830277060032417592098008847162727, 14068379226083299071248895931891435683229

Offset: 0

Alois P. Heinz, Feb 16 2014

nonn

From Gus Wiseman, May 31 2019: (Start)
Also the number of strict integer partitions of 2^n with n parts. For example, the a(1) = 1 through a(4) = 9 partitions are (A = 10):
  (2)  (31)  (431)  (6532)
             (521)  (6541)
                    (7432)
                    (7531)
                    (7621)
                    (8431)
                    (8521)
                    (9421)
                    (A321)
(End)

			a(1) = 1: 11.
a(2) = 1: 211.
a(3) = 2: 3221, 32111.
a(4) = 9: 433321, 443221, 4322221, 4332211, 4432111, 43222111, 43321111, 432211111, 4321111111.

Alois P. Heinz, Table of n, a(n) for n = 0..62
A. V. Sills and D. Zeilberger, Formulae for the number of partitions of n into at most m parts (using the quasi-polynomial ansatz), arXiv:1108.4391 [math.CO], 2011.

Column k=2 of A238012.
Cf. A236810, A237512, A237998, A238000, A238001.
Cf. A000009, A002033, A067735, A126796, A283111.

a[n_] := SeriesCoefficient[Product[1/(1 - x^j), {j, 1, n}], {x, 0, 2^n - n*(n + 1)/2}];
Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 15}] (* Jean-François Alcover, Aug 19 2018 *)

a(n) = [x^(2^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 2^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

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A238000 Number of partitions of n^n into parts that are at most n.

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A238012 Number A(n,k) of partitions of k^n into parts that are at most n with at least one part of each size; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A237998 Number of partitions of 2^n into parts that are at most n.

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A237999 Number of partitions of 2^n into parts that are at most n with at least one part of each size.

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Programs

Mathematica

Formula