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.

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A266692 Number of partitions of n with product of multiplicities of parts equal to 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 1, 3, 2, 4, 5, 7, 7, 14, 13, 18, 25, 29, 35, 47, 54, 65, 86, 101, 120, 147, 174, 205, 254, 291, 347, 419, 486, 565, 676, 779, 908, 1065, 1228, 1425, 1668, 1906, 2198, 2547, 2912, 3336, 3841, 4384, 4998, 5728, 6513, 7404, 8436
Offset: 0

Views

Author

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

Keywords

Examples

			a(9) = 2: [1,1,1,1,1,1,1,1,1], [1,1,1,2,2,2].
a(11) = 1: [1,1,1,1,1,1,1,1,1,2].
a(12) = 3: [1,1,1,1,1,1,1,1,1,3], [1,1,1,2,2,2,3], [1,1,1,3,3,3].

Links

Crossrefs

Column k=9 of A266477.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)0, 0, (h->
       b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
    end:
a:= b(n$2, 9):
seq(a(n), n=0..65);
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[i*(p + (i - 1)/2) < n, 0, If[n == 0, If[p == 1, 1, 0], b[n, i - 1, p] + Sum[If[Mod[p, j] > 0, 0, Function[h, b[h, Min[h, i - 1], p/j]][n - i*j]], {j, 1, Min[p, n/i]}]]];
a[n_] := b[n, n, 9];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, May 01 2018, translated from Maple *)

Formula

a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(1/4), where c = 0.004308121528... - Vaclav Kotesovec, May 24 2018

A266693 Number of partitions of n with product of multiplicities of parts equal to 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 4, 2, 7, 8, 12, 13, 19, 22, 32, 36, 46, 64, 72, 88, 112, 134, 160, 203, 236, 287, 343, 412, 477, 577, 676, 798, 944, 1101, 1283, 1516, 1754, 2030, 2361, 2738, 3157, 3657, 4202, 4826, 5567, 6356, 7279, 8340, 9494, 10815
Offset: 0

Views

Author

Emeric Deutsch and Alois P. Heinz, Jan 02 2016

Keywords

Examples

			a(9) = 1: [1,1,1,1,1,2,2].
a(12) = 3: [1,1,1,1,1,1,1,1,1,1,2], [1,1,2,2,2,2,2], [1,1,1,1,1,2,2,3].
a(13) = 4: [1,1,1,1,1,1,1,1,1,1,3], [1,1,1,1,1,2,3,3], [1,1,1,1,1,2,2,4], [1,1,1,1,1,4,4].

Links

Crossrefs

Column k=10 of A266477.

Programs

  • Maple
    b:= proc(n, i, p) option remember; `if`(i*(p+(i-1)/2)0, 0, (h->
       b(h, min(h, i-1), p/j))(n-i*j)), j=1..min(p, n/i))))
    end:
a:= b(n$2, 10):
seq(a(n), n=0..65);
  • Mathematica
    b[n_, i_, p_] := b[n, i, p] = If[i*(p + (i - 1)/2) < n, 0, If[n == 0, If[p == 1, 1, 0], b[n, i - 1, p] + Sum[If[Mod[p, j] > 0, 0, Function[h, b[h, Min[h, i - 1], p/j]][n - i*j]], {j, 1, Min[p, n/i]}]]];
a[n_] := b[n, n, 10];
Table[a[n], {n, 0, 65}] (* Jean-François Alcover, May 01 2018, translated from Maple *)

Formula

a(n) ~ c * exp(Pi*sqrt(n/3)) * n^(1/4), where c = 0.007782666499... - Vaclav Kotesovec, May 24 2018

A353504 Number of integer partitions of n whose product is less than the product of their multiplicities.

Original entry on oeis.org

0, 0, 1, 1, 1, 2, 2, 4, 4, 6, 6, 8, 9, 11, 12, 14, 14, 18, 21, 23, 26, 29, 29, 33, 36, 39, 40, 43, 44, 50, 53, 55, 59, 65, 69, 72, 78, 79, 81, 85, 92, 95, 97, 100, 103, 108, 109, 112, 118, 124, 129, 137, 139, 142, 149, 155, 159, 165, 166, 173, 178, 181, 187
Offset: 0

Views

Author

Gus Wiseman, May 19 2022

Keywords

Examples

			The a(2) = 1 through a(9) = 6 partitions:
  11   111   1111   2111    21111    22111     221111     222111
                    11111   111111   31111     311111     411111
                                     211111    2111111    2211111
                                     1111111   11111111   3111111
                                                          21111111
                                                          111111111

Crossrefs

LHS (product of parts) is counted by A339095, ranked by A003963.
RHS (product of multiplicities) is counted by A266477, ranked by A005361.
The version for greater instead of less is A353505.
The version for equal instead of less is A353506, ranked by A353503.
A181819 gives prime shadow, with an inverse A181821.
A353398 counts partitions with the same product of multiplicities as of shadows, ranked by A353399.
Cf. A002033, A008284, A085629, A097318, A098859, A114640, A116608, A118914, A124010, A304678, A353394, A353500, A353507.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Times@@#

A353505 Number of integer partitions of n whose product is greater than the product of their multiplicities.

Original entry on oeis.org

0, 0, 1, 2, 3, 5, 7, 11, 17, 24, 35, 47, 66, 89, 121, 162, 214, 276, 362, 464, 599, 763, 971, 1219, 1537, 1918, 2393, 2966, 3668, 4512, 5549, 6784, 8287, 10076, 12238, 14807, 17898, 21556, 25931, 31094, 37243, 44486, 53075, 63158, 75069, 89025, 105447, 124636
Offset: 0

Views

Author

Gus Wiseman, May 19 2022

Keywords

Examples

			The a(0) = 0 through a(7) = 11 partitions:
  .  .  (2)  (3)   (4)   (5)    (6)    (7)
             (21)  (22)  (32)   (33)   (43)
                   (31)  (41)   (42)   (52)
                         (221)  (51)   (61)
                         (311)  (222)  (322)
                                (321)  (331)
                                (411)  (421)
                                       (511)
                                       (2221)
                                       (3211)
                                       (4111)

Crossrefs

RHS (product of multiplicities) is counted by A266477, ranked by A005361.
LHS (product of parts) is counted by A339095, ranked by A003963.
The version for less instead of greater is A353504.
The version for equality is A353506, ranked by A353503.
A124010 gives prime signature, sorted A118914.
A181819 gives prime shadow, with an inverse A181821.
A353398 counts partitions with the same products of multiplicities as of shadows, ranked by A353399.
Cf. A002033, A008284, A008619, A085629, A097318, A098859, A114640, A116608, A130091, A304678, A353394, A353507.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],Times@@#>Times@@Length/@Split[#]&]],{n,0,30}]

A353741 Irregular triangle read by rows where T(n,k) is the number of integer partitions of n with product k, all zeros removed.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 3, 1, 1, 3, 1, 3, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 1, 3, 1, 1, 3, 2, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 3, 1, 1, 4, 2, 2, 1, 4, 1, 1, 1, 3, 2
Offset: 0

Views

Author

Gus Wiseman, May 20 2022

Keywords

Comments

Warning: There are certain internal "holes" in A339095 that are removed in this sequence.

Examples

			Triangle begins:
  1
  1
  1 1
  1 1 1
  1 1 1 2
  1 1 1 2 1 1
  1 1 1 2 1 2 2 1
  1 1 1 2 1 2 1 2 1 1 2
  1 1 1 2 1 2 1 3 1 1 3 1 3 1
  1 1 1 2 1 2 1 3 2 1 3 1 1 3 2 2 2 1
  1 1 1 2 1 2 1 3 2 2 3 1 1 4 2 2 1 4 1 1 1 3 2
Row n = 7 counts the following partitions:
  1111111   211111   31111   4111    511   61     7   421    331   52   43
                             22111         3211       2221              322

Crossrefs

Row sums are A000041.
Row lengths are A034891.
A partial transpose is A319000.
The full version with zeros is A339095, rank statistic A003963.
A008284 counts partitions by sum, strict A116608.
A225485 counts partitions by frequency depth.
A266477 counts partitions by product of multiplicities, ranked by A005361.
Cf. A002033, A266499, A325242, A325268, A325280, A353503, A353506, A353698.

Programs

  • Mathematica
    DeleteCases[Table[Length[Select[IntegerPartitions[n],Times@@#==k&]],{n,0,10},{k,1,2^n}],0,2]
Previous Showing 21-25 of 25 results.

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