A376369 -id:A376369 - cp's OEIS frontend

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).

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.

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

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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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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