A376369 -id:A376369 - cp's OEIS frontend

A376378 Indices of records in A376369.

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)).

A376663 Largest frequency of n in the multiset of multinomial coefficients k!/(x_1! * ... * x_j!) with 1 <= x_1 <= ... <= x_j for a fixed k = x_1 + ... + x_j, i.e., maximum number of times that n appears in a row of A036038.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

			56 appears twice in row 8 of A036038 (and never more than twice in the same row): 56 = 8!/(1!*1!*6!) = 8!/(3!*5!). Hence, a(56) = 2.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A036038, A376369, A376661, A376664, A376665 (records), A376666 (indices of records), A376667.

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).

Original entry on oeis.org

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.

A376367 Sorted multinomial coefficients greater than 1, including duplicates.

Original entry on oeis.org

2, 3, 4, 5, 6, 6, 6, 7, 8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 15, 16, 17, 18, 19, 20, 20, 20, 21, 21, 22, 23, 24, 24, 25, 26, 27, 28, 28, 29, 30, 30, 30, 31, 32, 33, 34, 35, 35, 36, 36, 37, 38, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 49, 50, 51, 52, 53

Offset: 1

Pontus von Brömssen, Sep 22 2024

nonn

Sorted terms of A036038, A050382, A078760, or A318762, excluding 1 (which appears infinitely often).

The number k appears A376369(k) times.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Cf. A036038, A050382, A078760, A318762, A376369, A376379.

a(n) = A318762(A376379(n)).

A376376 Least number that can be written as a multinomial coefficient in exactly n ways, or 0 if no such number exists.

Original entry on oeis.org

2, 10, 6, 420, 120, 210, 4324320, 7207200, 720720, 360360, 6983776800, 9777287520, 13967553600, 48886437600, 195545750400, 24736537425600, 586637251200, 293318625600, 148419224553600, 742096122768000, 28941748787952000, 296838449107200, 1736504927277120000

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

a(n) is the least number that occurs exactly n times in A036038 or A376367, i.e., the least number m such that A376369(m) = n.
After a(62), the sequence continues (where "?" represents terms that are either 0 or greater than 10^29): ?, 92098021748598694855458432000, ?, 6268725246643132945351680000, 1567181311660783236337920000, ?, 3134362623321566472675840000. After a(69), all terms are either 0 or larger than 10^29.
It seems that a(n) often is in A025487, at least for small n. The exceptions are n = 2, 12, 26, 30, 31, 33, 34, 35, 36, 37, 38, 42, 44, ... .

Pontus von Brömssen, Table of n, a(n) for n = 1..62

First column of A376370.

Cf. A025487, A036038, A376367, A376369.

A376661 Frequency of the most common number among the multinomial coefficients n!/(x_1! * ... * x_k!) for all partitions (x_1, ..., x_k) of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 4, 6, 6, 7, 8, 9, 11, 11, 13, 13, 14, 15, 16, 18, 19, 20, 23, 24, 26, 27, 30, 33, 37, 40, 43, 49, 52, 57, 64, 68, 76, 79, 87, 93, 99, 109, 116, 125, 135, 143, 157, 171, 191, 206, 223, 238, 254, 276, 291

Offset: 0

Pontus von Brömssen, Oct 02 2024

nonn

Frequency of the most common number in row n of A036038 (for n >= 1) or A078760.
The sequence is nondecreasing, because a set of partitions of n-1 with a common multinomial coefficient can be extended to a set of partitions of n with a common multinomial coefficient by adding a unit part to each partition. It appears that a(n) > a(n-1) for n >= 28.
The sequence is unbounded. To see this, note that the sets of parts (1,1,1,4) and (2,2,3) of a partition can be exchanged without affecting the value of the multinomial coefficient, because 1+1+1+4 = 2+2+3 and 1!*1!*1!*4! = 2!*2!*3!. In particular, a((7*k)!/24^k) >= k+1 from the partitions 7*k = (3*j)*1 + j*4 + (2*(k-j))*2 + (k-j)*3 for 0 <= j <= k.

			For n = 7, the only number that appears more than once in row 7 of A036038 is 210, which appears twice: 210 = 7!/(2!*2!*3!) = 7!/(1!*1!*1!*4!). Hence, a(7) = 2.

Cf. A036038, A070289, A078760, A376369, A376662, A376663.

A376371 Numbers that occur exactly once in A036038, i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!), with 1 <= x_1 <= ... <= x_k, is equal to m only when (x_1, ..., x_k) = (1, m-1).

Original entry on oeis.org

2, 3, 4, 5, 7, 8, 9, 11, 13, 14, 16, 17, 18, 19, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 37, 38, 39, 40, 41, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 73, 74, 75, 76, 77, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

Numbers m such that A376369(m) = 1, i.e., numbers that appear only once in A376367.

			10 is not a term, because it can be represented as a multinomial coefficient in 2 ways: 10 = 10!/(1!*9!) = 5!/(2!*3!).

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

First row of A376370.
Complement of A325472 (with respect to the positive integers).
Cf. A036038, A376367, A376369.

A376373 Numbers that occur exactly 3 times in A036038, i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly 3 integer partitions (x_1, ..., x_k).

Original entry on oeis.org

6, 20, 30, 56, 60, 90, 105, 252, 360, 462, 495, 504, 560, 720, 756, 990, 1320, 1365, 1540, 1716, 2970, 3360, 3960, 4290, 4620, 5460, 6006, 6435, 7920, 8190, 10080, 10296, 10626, 10920, 11628, 12012, 12870, 14280, 15504, 17550, 18360, 21840, 23256, 24024, 24310

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

Numbers m such that A376369(m) = 3, i.e., numbers that appear exactly 3 times in A376367.

			6 is a term, because it can be represented as a multinomial coefficient in exactly 3 ways: 6 = 6!/(1!*5!) = 4!/(2!*2!) = 3!/(1!*1!*1!).

Pontus von Brömssen, Table of n, a(n) for n = 1..10000

Third row of A376370.
Subsequence of A325593.
Cf. A036038, A376367, A376369.

A376368 Least number k with a partition k = x_1 + ... + x_j such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to n.

Original entry on oeis.org

0, 2, 3, 4, 5, 3, 7, 8, 9, 5, 11, 4, 13, 14, 6, 16, 17, 18, 19, 5, 7, 22, 23, 4, 25, 26, 27, 8, 29, 5, 31, 32, 33, 34, 7, 9, 37, 38, 39, 40, 41, 7, 43, 44, 10, 46, 47, 48, 49, 50, 51, 52, 53, 54, 11, 8, 57, 58, 59, 5, 61, 62, 63, 64, 65, 12, 67, 68, 69, 8, 71

Offset: 1

Pontus von Brömssen, Sep 22 2024

nonn

Index of first row of A078760 (or A036038 when n >= 2) that contains n.

a(n) <= n, with equality if and only if n is in A376371, i.e., if and only if n is not in A325472.

			a(6) = 3, because 6 appears in row 3 of A078760, corresponding to the multinomial coefficient 3!/(1!*1!*1!) = 6.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Pontus von Brömssen, Log-log plot, using Plot2.

Cf. A036038, A078760, A325472, A376369, A376370, A376371.

a(k!) = k for k != 1.

cp's OEIS Frontend

A376378 Indices of records in A376369.

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A376377 Records in A376369.

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A376663 Largest frequency of n in the multiset of multinomial coefficients k!/(x_1! * ... * x_j!) with 1 <= x_1 <= ... <= x_j for a fixed k = x_1 + ... + x_j, i.e., maximum number of times that n appears in a row of A036038.

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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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A376367 Sorted multinomial coefficients greater than 1, including duplicates.

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A376376 Least number that can be written as a multinomial coefficient in exactly n ways, or 0 if no such number exists.

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A376661 Frequency of the most common number among the multinomial coefficients n!/(x_1! * ... * x_k!) for all partitions (x_1, ..., x_k) of n.

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A376371 Numbers that occur exactly once in A036038, i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!), with 1 <= x_1 <= ... <= x_k, is equal to m only when (x_1, ..., x_k) = (1, m-1).

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A376373 Numbers that occur exactly 3 times in A036038, i.e., numbers m such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) is equal to m for exactly 3 integer partitions (x_1, ..., x_k).

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A376368 Least number k with a partition k = x_1 + ... + x_j such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to n.

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