cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

A238016 Number A(n,k) of partitions of n^k into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

0, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 5, 12, 5, 1, 1, 1, 9, 75, 64, 7, 1, 1, 1, 17, 588, 2280, 377, 11, 1, 1, 1, 33, 5043, 123464, 106852, 2432, 15, 1, 1, 1, 65, 44652, 7566280, 55567352, 6889527, 16475, 22, 1
Offset: 0

Views

Author

Alois P. Heinz, Feb 17 2014

Keywords

Comments

In general, for k>3, is column k asymptotic to exp(2*n) * n^((k-2)*n-k) / (2*Pi). For k=1 see A000041, for k=2 see A206226 and for k=3 see A238608. - Vaclav Kotesovec, May 25 2015
Conjecture: If f(n) >= O(n^4) then "number of partitions of f(n) into parts that are at most n" is asymptotic to f(n)^(n-1) / (n!*(n-1)!). See also A237998, A238000, A236810 or A258668-A258672. - Vaclav Kotesovec, Jun 07 2015

Examples

			A(3,1) = 3: 3, 21, 111.
A(3,2) = 12: 333, 3222, 3321, 22221, 32211, 33111, 222111, 321111, 2211111, 3111111, 21111111, 111111111.
A(2,3) = 5: 2222, 22211, 221111, 2111111, 11111111.
A(2,4) = 9: 22222222, 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111, 1111111111111111.
Square array A(n,k) begins:
  0, 1,   1,      1,        1,           1, ...
  1, 1,   1,      1,        1,           1, ...
  1, 2,   3,      5,        9,          17, ...
  1, 3,  12,     75,      588,        5043, ...
  1, 5,  64,   2280,   123464,     7566280, ...
  1, 7, 377, 106852, 55567352, 33432635477, ...

Links

Crossrefs

Columns k=0-10 give: A057427, A000041, A206226, A238608, A238609, A238610, A238611, A238612, A238613, A238614, A238615.
Rows n=0-10 give: A057427, A000012, A094373, A238630, A238631, A238632, A238633, A238634, A238635, A238636, A238637.
Main diagonal gives A238000.
Cf. A238010.

Programs

  • Mathematica
    A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, n^k}]; A[0, 0] = 0; Table[A[n-k, k], {n, 0, 10}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Oct 11 2015 *)

Formula

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

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

Original entry on oeis.org

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

Views

Author

Alois P. Heinz, Feb 16 2014

Keywords

Examples

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

Links

Crossrefs

Main diagonal of A238010 and A238016.
Cf. A236810, A237512, A237998, A237999, A238001.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Alois P. Heinz, Feb 16 2014

Keywords

Comments

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

Examples

			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, ...

Links

Crossrefs

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.

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Alois P. Heinz, Feb 16 2014

Keywords

Examples

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

Links

Crossrefs

Column k=2 of A238010.
Cf. A236810, A237512, A237999, A238000, A238001, A258672.

Programs

  • Mathematica
    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 *)

Formula

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

A258670 Number of partitions of (2*n)! into parts that are at most n.

Original entry on oeis.org

0, 1, 13, 43561, 455366036161, 60209252317216962943201, 291857679749953126623181556402787323521, 120972618144269517756284629487432992029777542693069847287041
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 07 2015

Keywords

Comments

Conjecture: If f(n) >= O(n^4) then "number of partitions of f(n) into parts that are at most n" is asymptotic to f(n)^(n-1) / (n!*(n-1)!). For the examples see A238016 and A238010.

Links

Crossrefs

Cf. A236810, A237998, A238000, A238010, A238016, A258668, A258669, A258671.

Formula

a(n) ~ (2*n)!^(n-1) / (n!*(n-1)!).

A258672 Number of partitions of n*2^n into parts that are at most n.

Original entry on oeis.org

0, 1, 5, 61, 2280, 273052, 110537709, 156456474138, 790541795804221, 14445283925963101577, 963056085414756870071490, 235864774408401842540220265704, 213426797830699546133563821747980513, 717147073290996884137625501875655000693923
Offset: 0

Views

Author

Vaclav Kotesovec, Jun 07 2015

Keywords

Comments

Conjecture: If f(n) >= O(n^4) then "number of partitions of f(n) into parts that are at most n" is asymptotic to f(n)^(n-1) / (n!*(n-1)!). For the examples see A238016 and A238010.

Links

Crossrefs

Cf. A236810, A237998, A238000, A238010, A238016.

Formula

a(n) ~ n^n * 2^(n*(n-1)) / (n!)^2.

A238560 Number of partitions of 3^n into parts that are at most n.

Original entry on oeis.org

0, 1, 5, 75, 4410, 1366617, 2559274110, 31328639384771, 2625213100478051111, 1553872467564223628517240, 6655897240266476140036201639917, 210488414263886836416720847147423569801, 49987740079047684574220644720678455290986424137
Offset: 0

Views

Author

Alois P. Heinz, Feb 28 2014

Keywords

Examples

			a(2) = 5: 22221, 222111, 2211111, 21111111, 111111111.

Links

Crossrefs

Column k=3 of A238010.

Formula

a(n) = [x^(3^n)] Product_{j=1..n} 1/(1-x^j).
a(n) ~ 3^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A238561 Number of partitions of 4^n into parts that are at most n.

Original entry on oeis.org

0, 1, 9, 374, 123464, 393073019, 13515852419746, 5357744226076852121, 25600195480450832892945051, 1525225328241455762364837330772150, 1164060788951887659290296574533366111395142, 11633609031659470387047660421170953987903118055988725
Offset: 0

Views

Author

Alois P. Heinz, Feb 28 2014

Keywords

Examples

			a(2) = 9: 22222222, 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111, 1111111111111111.

Links

Crossrefs

Column k=4 of A238010.

Formula

a(n) = [x^(4^n)] Product_{j=1..n} 1/(1-x^j).
a(n) ~ 4^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A238562 Number of partitions of 5^n into parts that are at most n.

Original entry on oeis.org

0, 1, 13, 1365, 1736385, 33432635477, 10815459920194632, 62725487942251841319705, 6831392910998237157682785667015, 14474273684384810126076369987535894403747, 613458701796516369003780850311157775345255117642867
Offset: 0

Views

Author

Alois P. Heinz, Feb 28 2014

Keywords

Examples

			a(2) = 13: 2222222222221, 22222222222111, 222222222211111, 2222222221111111, 22222222111111111, 222222211111111111, 2222221111111111111, 22222111111111111111, 222211111111111111111, 2221111111111111111111, 22111111111111111111111, 211111111111111111111111, 1111111111111111111111111.

Links

Crossrefs

Column k=5 of A238010.

Formula

a(n) = [x^(5^n)] Product_{j=1..n} 1/(1-x^j).
a(n) ~ 5^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A238563 Number of partitions of 6^n into parts that are at most n.

Original entry on oeis.org

0, 1, 19, 3997, 15292153, 1274403730688, 2561606354507677872, 132653831108423573746282961, 185588704806236441807500779350272919, 7271336250750488290453701705473754841288395525, 8205182525221704785195056768847594152799767482152756236799
Offset: 0

Views

Author

Alois P. Heinz, Feb 28 2014

Keywords

Links

Crossrefs

Column k=6 of A238010.

Formula

a(n) = [x^(6^n)] Product_{j=1..n} 1/(1-x^j).
a(n) ~ 6^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015
Showing 1-10 of 14 results. Next

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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