A376376 -id:A376376 - cp's OEIS frontend

A376370 Square array read by antidiagonals: row n lists numbers that occur exactly n times in A036038 (or A050382 or A078760 or A318762), i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly n integer partitions (x_1, ..., x_k).

2, 3, 10, 4, 12, 6, 5, 15, 20, 420, 7, 21, 30, 630, 120, 8, 24, 56, 840, 1680, 210, 9, 28, 60, 1980, 60060, 1260, 4324320, 11, 35, 90, 3003, 83160, 2520, 21621600, 7207200, 13, 36, 105, 7140, 180180, 5040, 24504480, 151351200, 720720

Offset: 1

Pontus von Brömssen, Sep 22 2024

Row n lists numbers m such that A376369(m) = n.
In case there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.
Any integer k >= 2 appears exactly once in the array.

			Array begins:
  n\k|       1         2         3         4         5          6          7          8
  ---+---------------------------------------------------------------------------------
  1  |       2         3         4         5         7          8          9         11
  2  |      10        12        15        21        24         28         35         36
  3  |       6        20        30        56        60         90        105        252
  4  |     420       630       840      1980      3003       7140       7560       9240
  5  |     120      1680     60060     83160    180180     240240     831600     900900
  6  |     210      1260      2520      5040     27720     166320    1441440    4084080
  7  | 4324320  21621600  24504480  43243200  75675600  116396280  367567200  908107200
  8  | 7207200 151351200 302702400 411863760 823727520 1816214400 2327925600 4655851200

Pontus von Brömssen, Table of n, a(n) for n = 1..1081 (antidiagonals 1..46)

Cf. A036038, A050382, A078760, A318762, A325472 (complement of first row), A325593 (complement of the union of the first 2 rows), A376369, A376376 (first column).

First five rows are A376371, A376372, A376373, A376374, A376375.

A376673 Least number whose maximum frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., least number m such that A376663(m) = n, or 0 if no such number exists.

Original entry on oeis.org

1, 56, 166320, 4084080, 1396755360, 698377680, 146659312800, 1075501627200, 37104806138400, 3710480613840000, 296838449107200, 86825246363856000, 96472495959840000, 36466603472819520000, 35251050023725536000, 272194921062320256000, 408292381593480384000

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

After a(36), the sequence continues (where "?" represents terms that are either 0 or greater than 10^29): ?, 3059734941813910128088320000, ?, ?, 64254433778092112689854720000. After a(41), all terms are either 0 or greater than 10^29.

The terms a(1), a(3), ..., a(15), a(24), a(26), ..., a(36), a(38), a(41) are all in A025487, but a(16), ..., a(23), a(25) are all divisible by 17^2 but not by 13^2.

			First few terms and their representations as multinomial coefficients (corresponding to partitions with sum A376664(n)):
  a(1) =          1 = 0!;
  a(2) =         56 = 8!/(1!*1!*6!) = 8!/(3!*5!);
  a(3) =     166320 = 12!/(1!*1!*1!*4!*5!) = 12!/(1!*1!*2!*2!*6!) = 12!/(2!*2!*3!*5!);
  a(4) =    4084080 = 17!/(1!*1!*1!*4!*10!) = 17!/(1!*2!*5!*9!) = 17!/(2!*2!*3!*10!) = 17!/(4!*6!*7!);
  a(5) = 1396755360 = 19!/(1!*1!*1!*1!*1!*4!*10!) = 19!/(1!*1!*1!*2!*5!*9!) = 19!/(1!*1!*2!*2!*3!*10!) = 19!/(1!*1!*4!*6!*7!) = 19!/(3!*4!*5!*7!).

Pontus von Brömssen, Table of n, a(n) for n = 1..36

First column of A376667.

Cf. A025487, A036038, A078760, A376376, A376663, A376664.

A376378 Indices of records in A376369.

Original entry on oeis.org

2, 6, 120, 210, 360360, 6983776800, 9777287520, 13967553600, 48886437600, 195545750400, 293318625600, 148419224553600, 296838449107200, 868252463638560000, 4002866486210592000, 8005732972421184000, 68048730265580064000, 136097460531160128000, 4082923815934803840000

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

Pontus von Brömssen, Table of n, a(n) for n = 1..28

Cf. A376369, A376376, A376377.

a(n) = A376376(A376377(n)).

A376377 Records in A376369.

Original entry on oeis.org

1, 3, 5, 6, 10, 11, 12, 13, 14, 15, 18, 19, 22, 24, 25, 28, 31, 37, 44, 50, 51, 52, 57, 59, 60, 62, 67, 69

Offset: 1

Pontus von Brömssen, Sep 23 2024

Cf. A376369, A376376, A376378.

a(n) = A376369(A376378(n)).

A376866 a(n) = smallest integer k >= 2 such that there exist n disjoint multisets of positive integers, whose corresponding multinomial coefficients equal k.

Original entry on oeis.org

2, 6, 120, 3003, 433866230594439538896000

Offset: 1

Pontus von Brömssen, Oct 07 2024

			  n |                     a(n) | disjoint multisets with multinomial coefficient a(n)
  --+--------------------------+---------------------------------------------
  1 |                        2 | {1,1}
  2 |                        6 | {2,2}, {1,5}
  3 |                      120 | {3,7}, {2,14}, {1,119}
  4 |                     3003 | {6,8}, {5,10}, {2,76}, {1,3002}
  5 | 433866230594439538896000 | {4,9,13,23}, {2,2,8,11,27}, {3,3,3,16,25}, {5,6,15,24},
    |                          | {1,433866230594439538895999}

Cf. A180058, A376369, A376376, A376828.

Edited by Max Alekseyev, May 18 2025

cp's OEIS Frontend

A376370 Square array read by antidiagonals: row n lists numbers that occur exactly n times in A036038 (or A050382 or A078760 or A318762), i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly n integer partitions (x_1, ..., x_k).

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A376673 Least number whose maximum frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., least number m such that A376663(m) = n, or 0 if no such number exists.

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A376378 Indices of records in A376369.

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A376377 Records in A376369.

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A376866 a(n) = smallest integer k >= 2 such that there exist n disjoint multisets of positive integers, whose corresponding multinomial coefficients equal k.

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