A325593 -id:A325593 - 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.

A325472 Numbers having at least two representations as multinomial coefficients M(n;lambda), where lambda is a partition of n.

Original entry on oeis.org

1, 6, 10, 12, 15, 20, 21, 24, 28, 30, 35, 36, 42, 45, 55, 56, 60, 66, 70, 72, 78, 84, 90, 91, 105, 110, 120, 126, 132, 136, 140, 153, 156, 165, 168, 171, 180, 182, 190, 210, 220, 231, 240, 252, 253, 272, 276, 280, 286, 300, 306, 325, 330, 336, 342, 351, 360

Offset: 1

Alois P. Heinz, Sep 06 2019

nonn

Numbers that are repeated in the triangle A036038 (all positive integers occur at least once).

All triangular numbers (A000217) except 0 and 3 are in this sequence.

			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.
42 is in the sequence because M(42;41,1) = M(7;5,1,1) = 42.

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Cf. A000041, A000217, A036038, A305188, A325306, A325593, A325901.

a(n) = A305188(n-1) for n > 1.

A325903 Numbers having at least three representations as multinomial coefficients M(n;lambda), where lambda is a partition of n into distinct parts.

Original entry on oeis.org

1, 105, 120, 210, 495, 1260, 1365, 1540, 3003, 4620, 5460, 6435, 7140, 10296, 11628, 15504, 24310, 27720, 29260, 30030, 42504, 43680, 45045, 77520, 83160, 102960, 116280, 120120, 180180, 203490, 352716, 360360, 376740, 437580, 593775, 657800, 680680, 720720

Offset: 1

Alois P. Heinz, Sep 07 2019

nonn

Numbers occurring at least three times in the triangle A309992.

All terms are contained in A325593 and in A325901.

			1 is in the sequence because M(0;0) = M(1;1) = M(2;2) = M(3;3) = ... = 1.
105 is in the sequence because M(7;4,2,1) = M(15;13,2) = M(105;104,1) = 105.
120 is in the sequence because M(10;7,3) = M(16;14,2) = M(120;119,1) = 120.
1365 is in the sequence because M(15;11,4) = M(15;12,2,1) = M(1365;1364,1) = 1365.

Alois P. Heinz, Table of n, a(n) for n = 1..318
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Cf. A000009, A309992, A325593, A325901.

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.

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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A325472 Numbers having at least two representations as multinomial coefficients M(n;lambda), where lambda is a partition of n.

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A325903 Numbers having at least three representations as multinomial coefficients M(n;lambda), where lambda is a partition of n into distinct parts.

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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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