A376370 -id:A376370 - cp's OEIS frontend

A376369 Number of nondecreasing tuples (x_1, ..., x_k) of positive integers (or integer partitions) such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) equals n.

1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 2

Pontus von Brömssen, Sep 22 2024

nonn

a(n) is the number of occurrences of n in each of A036038, A050382, A078760, A318762, and A376367.

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.

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

Pontus von Brömssen, Table of n, a(n) for n = 2..10000

Cf. A036038, A050382, A078760, A318762, A376367, A376368, A376370.

A376667 Square array read by antidiagonals: row n lists numbers whose maximal frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., numbers m such that A376663(m) = n.

Original entry on oeis.org

1, 2, 56, 3, 210, 166320, 4, 504, 360360, 4084080, 5, 1260, 720720, 17907120, 1396755360, 6, 1365, 2162160, 73513440, 4190266080, 698377680, 7, 1680, 5045040, 75675600, 4655851200, 13967553600, 146659312800, 8, 1716, 5765760, 220540320, 4942365120, 27935107200, 293318625600, 1075501627200

Offset: 1

Pontus von Brömssen, Oct 02 2024

In case there are only finitely many solutions for a certain value of n, the rest of that row is filled with 0's.

Each positive integer appears exactly once in the array, so as a linear sequence it is a permutation of the positive integers (unless there are any 0's).

			Array begins:
  n\k|             1             2              3              4              5              6
  ---+----------------------------------------------------------------------------------------
  1  |             1             2              3              4              5              6
  2  |            56           210            504           1260           1365           1680
  3  |        166320        360360         720720        2162160        5045040        5765760
  4  |       4084080      17907120       73513440       75675600      220540320      411863760
  5  |    1396755360    4190266080     4655851200     4942365120     9884730240    24443218800
  6  |     698377680   13967553600    27935107200   267711444000   537750813600   586637251200
  7  |  146659312800  293318625600  1606268664000  3226504881600  6184134356400  7228208988000
  8  | 1075501627200 6453009763200 12368268712800 24736537425600 29683844910720 74209612276800

Pontus von Brömssen, Table of n, a(n) for n = 1..465 (antidiagonals 1..30)

Cf. A036038, A078760, A325306 (complement of first row), A376370, A376663, A376673 (first column).

First five rows are A376668, A376669, A376670, A376671, A376672.

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.

A325593 Numbers having at least three representations as multinomial coefficients M(n;lambda), where lambda is a partition of n.

Original entry on oeis.org

1, 6, 20, 30, 56, 60, 90, 105, 120, 210, 252, 360, 420, 462, 495, 504, 560, 630, 720, 756, 840, 990, 1260, 1320, 1365, 1540, 1680, 1716, 1980, 2520, 2970, 3003, 3360, 3960, 4290, 4620, 5040, 5460, 6006, 6435, 7140, 7560, 7920, 8190, 9240, 10080, 10296, 10626

Offset: 1

Alois P. Heinz, Sep 06 2019

nonn

Numbers occurring at least three times in the triangle A036038.

			1 is in the sequence because M(0;0) = M(1;1) = M(2;2) = M(3;3) = ... = 1.
6 is in the sequence because M(6;5,1) = M(4;2,2) = M(3;1,1,1) = 6.
56 is in the sequence because M(56;55,1) = M(8;5,3) = M(8;6,1,1) = 56.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000 (first 505 terms from Alois P. Heinz)

Cf. A036038, A325472, A325903, A376370.

A376373 is a subsequence.

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.

A376372 Numbers that occur exactly twice 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 2 integer partitions (x_1, ..., x_k).

Original entry on oeis.org

10, 12, 15, 21, 24, 28, 35, 36, 42, 45, 55, 66, 70, 72, 78, 84, 91, 110, 126, 132, 136, 140, 153, 156, 165, 168, 171, 180, 182, 190, 220, 231, 240, 253, 272, 276, 280, 286, 300, 306, 325, 330, 336, 342, 351, 364, 378, 380, 406, 435, 455, 465, 496, 506, 528, 552

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

Numbers m such that A376369(m) = 2, i.e., numbers that appear exactly twice in A376367.

			10 is a term, because it can be represented as a multinomial coefficient in exactly 2 ways: 10 = 10!/(1!*9!) = 5!/(2!*3!).

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

Second row of A376370.
Subsequence of A325472.
Cf. A036038, A376367, A376369.

A376374 Numbers that occur exactly 4 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 4 integer partitions (x_1, ..., x_k).

Original entry on oeis.org

420, 630, 840, 1980, 3003, 7140, 7560, 9240, 13860, 15120, 25200, 43680, 53130, 55440, 72072, 90090, 116280, 120120, 142506, 277200, 278256, 332640, 371280, 415800, 450450, 480480, 813960, 1113840, 1261260, 1801800, 2018940, 2441880, 2702700, 3255840, 3326400

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

Numbers m such that A376369(m) = 4, i.e., numbers that appear exactly 4 times in A376367.

			420 is a term, because it can be represented as a multinomial coefficient in exactly 4 ways: 420 = 420!/(1!*419!) = 21!/(1!*1!*19!) = 8!/(2!*2!*4!) = 7!/(1!*1!*2!*3!).

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

Fourth row of A376370.

Cf. A036038, A376367, A376369.

A376375 Numbers that occur exactly 5 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 5 integer partitions (x_1, ..., x_k).

Original entry on oeis.org

120, 1680, 60060, 83160, 180180, 240240, 831600, 900900, 1081080, 1627920, 1663200, 2522520, 2882880, 3603600, 7567560, 10090080, 14414400, 20180160, 25225200, 30270240, 35814240, 36756720, 37837800, 46558512, 49008960, 51482970, 60540480, 61261200, 64864800

Offset: 1

Pontus von Brömssen, Sep 23 2024

nonn

Numbers m such that A376369(m) = 5, i.e., numbers that appear exactly 5 times in A376367.

			120 is a term, because it can be represented as a multinomial coefficient in exactly 5 ways: 120 = 120!/(1!*119!) = 16!/(2!*14!) = 10!/(3!*7!) = 6!/(1!*1!*1!*3!) = 5!/(1!*1!*1!*1!*1).

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

Fifth row of A376370.

Cf. A036038, A376367, A376369.

cp's OEIS Frontend

A376369 Number of nondecreasing tuples (x_1, ..., x_k) of positive integers (or integer partitions) such that the multinomial coefficient (x_1 + ... + x_k)!/(x_1! * ... * x_k!) equals n.

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A376667 Square array read by antidiagonals: row n lists numbers whose maximal frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., numbers m such that A376663(m) = n.

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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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A325593 Numbers having at least three representations as multinomial coefficients M(n;lambda), where lambda is a partition 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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A376372 Numbers that occur exactly twice 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 2 integer partitions (x_1, ..., x_k).

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A376374 Numbers that occur exactly 4 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 4 integer partitions (x_1, ..., x_k).

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A376375 Numbers that occur exactly 5 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 5 integer partitions (x_1, ..., x_k).

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