A238012 -id:A238012 - cp's OEIS frontend

A238010 Number A(n,k) of partitions of k^n into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 1, 1, 0, 1, 5, 10, 1, 1, 0, 1, 9, 75, 64, 1, 1, 0, 1, 13, 374, 4410, 831, 1, 1, 0, 1, 19, 1365, 123464, 1366617, 26207, 1, 1, 0, 1, 25, 3997, 1736385, 393073019, 2559274110, 2239706, 1, 1

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

			A(3,2) = 10: 332, 2222, 3221, 3311, 22211, 32111, 221111, 311111, 2111111, 11111111.
A(2,3) = 5: 22221, 222111, 2211111, 21111111, 111111111.
A(2,4) = 9: 22222222, 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111, 1111111111111111.
Square array A(n,k) begins:
  0, 0,   0,       0,         0,           0, ...
  1, 1,   1,       1,         1,           1, ...
  1, 1,   3,       5,         9,          13, ...
  1, 1,  10,      75,       374,        1365, ...
  1, 1,  64,    4410,    123464,     1736385, ...
  1, 1, 831, 1366617, 393073019, 33432635477, ...

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])

Columns k=0+1,2-10 give: A057427, A237998, A238560, A238561, A238562, A238563, A238564, A238565, A238566, A238567.
Rows n=0-2 give: A000004, A000012, A080827.
Main diagonal gives A238000.
Cf. A238012, A238016.

A[n_, k_] := SeriesCoefficient[Product[1/(1-x^j), {j, 1, n}], {x, 0, k^n}]; A[0, 0] = 0; Table[A[n-k, k], {n, 0, 9}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 17 2017 *)

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

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

A238001 Number of partitions of n^n into parts that are at most n with at least one part of each size.

Original entry on oeis.org

0, 1, 1, 48, 109809, 32796849930, 2555847904495965819, 85962759806610904434664386174, 1841132100297745277187328924904656111127054, 34687813181057391872792859998288408847592250236051615502024

Offset: 0

Alois P. Heinz, Feb 16 2014

nonn

			a(1) = 1: 1.
a(2) = 1: 211.
a(3) = 48: 3333333321, ..., 321111111111111111111111.

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], 2011.

Main diagonal of A238012.

Cf. A236810, A237512, A237998, A237999, A238000.

maxExponent = 50; a[0] = 0; a[1] = 1;
a[n_] := Module[{}, aparts = List @@ (Product[1/(1 - x^j), {j, 1, n}] // Apart); cc = aparts + O[x]^maxExponent // CoefficientList[#, x]&; f[k_] = Total[FindSequenceFunction[#, k]& /@ cc]; f[n^n-n(n+1)/2 + 1] // Round];
Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 9}] (* Jean-François Alcover, Nov 15 2018 *)

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

a(n) ~ n^(n*(n-1)) / (n!*(n-1)!) ~ exp(2*n) * n^(n*(n-3)) / (2*Pi). - Vaclav Kotesovec, Jun 05 2015

A239168 Number of partitions of 9^n into parts that are at most n with at least one part of each size.

Original entry on oeis.org

0, 1, 40, 43923, 1956835062, 4219267293723828, 490589938553810921101750, 3299246284983094033572923631218500, 1347808520417651710823757078029174789058075682, 34687813181057391872792859998288408847592250236051615502024

Offset: 0

Alois P. Heinz, Mar 11 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..34
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=9 of A238012.

a(n) = [x^(9^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 9^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A239162 Number of partitions of 3^n into parts that are at most n with at least one part of each size.

Original entry on oeis.org

0, 1, 4, 48, 3042, 1067474, 2215932130, 29012104252380, 2526293243761311036, 1525710603023191548743988, 6600334932211428773703751221040, 209705652574790086852527310591449309624, 49907101066058865036206450041083799915221352487

Offset: 0

Alois P. Heinz, Mar 11 2014

nonn

			a(2) = 4: 22221, 222111, 2211111, 21111111.

Alois P. Heinz, Table of n, a(n) for n = 0..49
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=3 of A238012.

maxExponent = 30; a[0] = 0; a[1] = 1;
a[n_] := Module[{}, aparts = List @@ (Product[1/(1 - x^j), {j, 1, n}] // Apart); cc = aparts + O[x]^maxExponent // CoefficientList[#, x]&; f[k_] = Total[FindSequenceFunction[#, k]& /@ cc]; f[3^n - n(n+1)/2 + 1] // Round]; Table[an = a[n];
Print[n, " ", an]; an, {n, 0, 12}] (* Jean-François Alcover, Nov 15 2018 *)

a(n) = [x^(3^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 3^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A239163 Number of partitions of 4^n into parts that are at most n with at least one part of each size.

Original entry on oeis.org

0, 1, 7, 310, 109809, 370702459, 13173778523786, 5303087097326728307, 25501946239758780918956349, 1523132187565775833398409415522245, 1163511401871888391788752975911167467265905, 11631778554448496258128131825307023131265496068454454

Offset: 0

Alois P. Heinz, Mar 11 2014

nonn

			a(2) = 7: 222222211, 2222221111, 22222111111, 222211111111, 2221111111111, 22111111111111, 211111111111111.

Alois P. Heinz, Table of n, a(n) for n = 0..43
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=4 of A238012.

maxExponent = 40; a[0] = 0; a[1] = 1;
a[n_] := Module[{}, aparts = List @@ (Product[1/(1 - x^j), {j, 1, n}] // Apart); cc = aparts + O[x]^maxExponent // CoefficientList[#, x]&; f[k_] = Total[FindSequenceFunction[#, k]& /@ cc]; f[4^n-n(n+1)/2 + 1] // Round];
Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 11}] (* Jean-François Alcover, Nov 15 2018 *)

a(n) = [x^(4^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 4^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A239164 Number of partitions of 5^n into parts that are at most n with at least one part of each size.

Original entry on oeis.org

0, 1, 12, 1240, 1655004, 32796849930, 10743023668660275, 62590747974586286694030, 6826987264035710020018176749475, 14471606032117455546329821353159274382372, 613427607589897771307393494301176209875530879140211

Offset: 0

Alois P. Heinz, Mar 11 2014

nonn

			a(2) = 12: 2222222222221, 22222222222111, 222222222211111, 2222222221111111, 22222222111111111, 222222211111111111, 2222221111111111111, 22222111111111111111, 222211111111111111111, 2221111111111111111111, 22111111111111111111111, 211111111111111111111111.

Alois P. Heinz, Table of n, a(n) for n = 0..40
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=5 of A238012.

maxExponent = 45; a[0] = 0; a[1] = 1;
a[n_] := Module[{}, aparts = List @@ (Product[1/(1 - x^j), {j, 1, n}] // Apart); cc = aparts + O[x]^maxExponent // CoefficientList[#, x]&; f[k_] = Total[FindSequenceFunction[#, k]& /@ cc];f[5^n - n(n+1)/2 + 1] // Round];
Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 10}] (* Jean-François Alcover, Nov 15 2018 *)

a(n) = [x^(5^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 5^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A239165 Number of partitions of 6^n into parts that are at most n with at least one part of each size.

Original entry on oeis.org

0, 1, 17, 3781, 14942231, 1264608203048, 2555847904495965819, 132574244496779071303074376, 185560862224740635595130202984468935, 7271076505438083132065943012753686950455454055, 8205115354631567886718289443554629632451344416164686337673

Offset: 0

Alois P. Heinz, Mar 11 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..37
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=6 of A238012.

maxExponent = 50; a[0] = 0; a[1] = 1;
a[n_] := Module[{}, aparts = List @@ (Product[1/(1 - x^j), {j, 1, n}] // Apart); cc = aparts + O[x]^maxExponent // CoefficientList[#, x]&; f[k_] = Total[FindSequenceFunction[#, k]& /@ cc]; f[6^n-n(n+1)/2+1] // Round];
Table[an = a[n]; Print[n, " ", an]; an, {n, 0, 10}] (* Jean-François Alcover, Nov 15 2018 *)

a(n) = [x^(6^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 6^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A239166 Number of partitions of 7^n into parts that are at most n with at least one part of each size.

Original entry on oeis.org

0, 1, 24, 9633, 95520600, 27656224652420, 260755601247189041231, 85962759806610904434664386174, 1041189281477724923668568740931602845066, 480588514551700434552887677121496205669535589365780, 8695551969224574889031840216144104978715552114947924501069394617

Offset: 0

Alois P. Heinz, Mar 11 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..36
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=7 of A238012.

a(n) = [x^(7^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 7^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

A239167 Number of partitions of 8^n into parts that are at most n with at least one part of each size.

Original entry on oeis.org

0, 1, 31, 21590, 475473009, 399953578562811, 14325140434481169064975, 23442235543128214521886383970201, 1841132100297745277187328924904656111127054, 7197719612276659958196058354497691622150052900765626132

Offset: 0

Alois P. Heinz, Mar 11 2014

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..35
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=8 of A238012.

a(n) = [x^(8^n-n*(n+1)/2)] Product_{j=1..n} 1/(1-x^j).

a(n) ~ 8^(n*(n-1)) / (n!*(n-1)!). - Vaclav Kotesovec, Jun 05 2015

cp's OEIS Frontend

A238010 Number A(n,k) of partitions of k^n into parts that are at most n; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A238001 Number of partitions of n^n into parts that are at most n with at least one part of each size.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A239168 Number of partitions of 9^n into parts that are at most n with at least one part of each size.

Views

Author

Keywords

Links

Crossrefs

Formula

A239162 Number of partitions of 3^n into parts that are at most n with at least one part of each size.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A239163 Number of partitions of 4^n into parts that are at most n with at least one part of each size.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A239164 Number of partitions of 5^n into parts that are at most n with at least one part of each size.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A239165 Number of partitions of 6^n into parts that are at most n with at least one part of each size.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A239166 Number of partitions of 7^n into parts that are at most n with at least one part of each size.