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 1-5 of 5 results.

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

A325830 Number of integer partitions of 2*n having exactly 2*n submultisets.

Original entry on oeis.org

0, 1, 1, 1, 3, 1, 10, 1, 21, 12, 15, 1, 121, 1, 20, 37, 309, 1, 319, 1, 309, 47, 33, 1, 3435, 30, 38, 405, 593, 1, 1574, 1, 11511, 80, 51, 77, 17552, 1, 56, 92, 13921, 1, 3060, 1, 1439, 2911, 69, 1, 234969, 56, 2044, 126, 1998, 1, 46488, 114, 36615, 137, 87, 1, 141906
Offset: 0

Views

Author

Gus Wiseman, May 25 2019

Keywords

Comments

If n is odd, there are no integer partitions of n with exactly n submultisets, so this sequence gives only the even-indexed terms.
The number of submultisets of an integer partition is the product of its multiplicities, each plus one.
The Heinz numbers of these partitions are given by A325793.

Examples

			The 12 submultisets of the partition (7221) are (), (1), (2), (7), (21), (22), (71), (72), (221), (721), (722), (7221), so (7221) is counted under a(6).
The a(1) = 1 through a(8) = 21 partitions (A = 10, B = 11):
  (2)  (31)  (411)  (431)   (61111)  (4332)    (8111111)  (6532)
                    (521)            (4431)               (6541)
                    (5111)           (5322)               (7432)
                                     (5331)               (7531)
                                     (6411)               (7621)
                                     (7221)               (8431)
                                     (7311)               (8521)
                                     (8211)               (9421)
                                     (33222)              (A321)
                                     (711111)             (44431)
                                                          (53332)
                                                          (63331)
                                                          (64222)
                                                          (73222)
                                                          (76111)
                                                          (85111)
                                                          (92221)
                                                          (94111)
                                                          (A3111)
                                                          (B2111)
                                                          (91111111)

Links

Crossrefs

Cf. A002033, A098859, A108917, A126796, A237999, A325694, A325792, A325793, A325828, A325831, A325832, A325833, A325834, A325836.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
      `if`(n=p-1, 1, 0), add(`if`(irem(p, j+1, 'r')=0,
      (w-> b(w, min(w, i-1), r))(n-i*j), 0), j=0..n/i))
    end:
a:= n-> `if`(isprime(n), 1, b(2*n$3)):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 16 2019
  • Mathematica
    Table[Length[Select[IntegerPartitions[2*n],Times@@(1+Length/@Split[#])==2*n&]],{n,0,30}]
(* Second program: *)
b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1,
     If[n == p - 1, 1, 0], Sum[If[Mod[p, j + 1] == 0, r = p/(j + 1);
     Function[w, b[w, Min[w, i - 1], r]][n - i*j], 0], {j, 0, n/i}]];
a[n_] := If[PrimeQ[n], 1, b[2n, 2n, 2n]];
a /@ Range[0, 60] (* Jean-François Alcover, May 12 2021, after Alois P. Heinz *)
  • PARI
    a(n)={if(n<1, 0, my(v=vector(2*n+1, k, vector(2*n))); v[1][1]=1; for(k=1, 2*n, forstep(j=#v, k, -1, for(m=1, (j-1)\k, for(i=1, 2*n\(m+1), v[j][i*(m+1)] += v[j-m*k][i])))); v[#v][2*n])} \\ Andrew Howroyd, Aug 16 2019

Formula

a(p) = 1 for prime p. - Andrew Howroyd, Aug 16 2019

Extensions

Terms a(31) and beyond from Andrew Howroyd, Aug 16 2019

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

Views

Author

Alois P. Heinz, Feb 16 2014

Keywords

Examples

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

Links

Crossrefs

Main diagonal of A238012.
Cf. A236810, A237512, A237998, A237999, A238000.

Programs

  • Mathematica
    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 *)

Formula

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
Showing 1-5 of 5 results.

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

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