A036038 -id:A036038 - 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.

A376664 Least number k such that there are A376663(n) partitions x_1 + ... + x_j = k such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to n, i.e., the first row k of A036038 in which n appears A376663(n) times (or 0 if n = 0).

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, Oct 02 2024

nonn

Differs from A376368 at n = 1260, 2520, 5040, 60060, 83160, ... . For example, 1260 appears first in row A376368(1260) = 7 of A036038, but only once. It also appears once in row 9, but in row a(1260) = 10 it appears A376663(1260) = 2 times.

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

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

Cf. A036038, A325472, A376368, A376663.

A215911 G.f.: exp( Sum_{n>=1} A215910(n)*x^n/n ), where A215910(n) equals the sum of the n-th power of multinomial coefficients in row n of triangle A036038.

Original entry on oeis.org

1, 1, 3, 84, 88602, 5137769389, 23588076629522583, 11893878960703225919597767, 876545054865944028047877165082786426, 12147135901759930712215268630715086378214795245696, 39632791164678725520866813137932593902239710762044280903318659253

Offset: 0

Paul D. Hanna, Aug 26 2012

nonn

			G.f.: A(x) = 1 + x + 3*x^2 + 84*x^3 + 88602*x^4 + 5137769389*x^5 +...
such that the logarithm of the g.f. begins:
log(A(x)) = x + 5*x^2/2 + 244*x^3/3 + 354065*x^4/4 + 25688403126*x^5/5 + 141528428949437282*x^6/6 +...+ A215910(n)*x^n/n +...
where the coefficients A215910(n) begin:
A215910(1) = 1^1 = 1;
A215910(2) = 1^2 + 2^2 = 5;
A215910(3) = 1^3 + 3^3 + 6^3 = 244;
A215910(4) = 1^4 + 4^4 + 6^4 + 12^4 + 24^4 = 354065;
A215910(5) = 1^5 + 5^5 + 10^5 + 20^5 + 30^5 + 60^5 + 120^5 = 25688403126; ...
and equal the sums of the n-th power of multinomial coefficients in row n of triangle A036038.

Vaclav Kotesovec, Table of n, a(n) for n = 0..30

Cf. A215910, A183239, A183241, A183241, A182963.

{a(n)=local(L=sum(m=1,n,m!^m*polcoeff(1/prod(k=1, n, 1-x^k/k!^m +x*O(x^m)), m)*x^m/m)+x*O(x^n));polcoeff(exp(L),n)}
for(n=0,15,print1(a(n),", "))

a(n) ~ (n!)^n / n. - Vaclav Kotesovec, Feb 19 2015

a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2 - 1) / exp(n^2 - 1/12). - Vaclav Kotesovec, Feb 19 2015

A257468 Triangle read by rows in which the n-th row lists the multinomials A036038 for all partitions of 2n with only even parts in Abramowitz-Stegun ordering.

Original entry on oeis.org

1, 1, 6, 1, 15, 90, 1, 28, 70, 420, 2520, 1, 45, 210, 1260, 3150, 18900, 113400, 1, 66, 495, 924, 2970, 13860, 34650, 83160, 207900, 1247400, 7484400, 1, 91, 1001, 3003, 6006, 45045, 84084, 210210, 270270, 1261260, 3153150, 7567560, 18918900, 113513400, 681080400

Offset: 1

Hartmut F. W. Hoft, Apr 25 2015

The row length sequence is A000041(n).
The triangle representation of this sequence has the same structure as the triangles in A036036 and A115621.
These multinomials, called M_1 by Abramowitz-Stegun on p. 831, are given in A036038.

			The first six rows of the irregular triangle. The columns headings indicate the number of parts in the underlying partitions. Brackets group the multinomials for all partitions of the same length m when there is more than one partition.
n\m 1        2             3        4        5
1:  1
2:  1        6
3:  1       15            90
4:  1  [28  70]          420     2520
5:  1  [45 210]   [1260 3150]   18900   113400
...
n = 6:  1 [66 495 924] [2970 13860 34650] [83160 207900] 1247400  7484400

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Cf. A000041, A036036, A036038, A115621.

(* row[] and triangle[] compute structured rows of the triangle *)
row[n_] := Map[Apply[Plus, #]!/Apply[Times, Map[Factorial, #]]&, GatherBy[2*IntegerPartitions[n], Length], {2}]
triangle[n_] := Map[row, Range[n]]
a[n_] := Flatten[triangle[n]]
a[7] (* data *)

Edited. - Wolfdieter Lang, May 09 2015

A376668 Positive integers that do not appear more than once in the same row of A036038 (or A078760), i.e., numbers m such that A376663(m) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

Is this the same as A357759? - R. J. Mathar, Oct 09 2024. [Answer: No, they are different. - Andrew Howroyd, Oct 09 2024]

			56 is not a term, because it can be represented as a multinomial coefficient for 2 different partitions of 8: 56 = 8!/(1!*1!*6!) = 8!/(3!*5!).

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

First row of A376667.
Complement of A325306 (with respect to the positive integers).
Cf. A036038, A078760, A376371, A376663.

A376669 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 2, i.e., numbers m such that A376663(m) = 2.

Original entry on oeis.org

56, 210, 504, 1260, 1365, 1680, 1716, 2520, 5040, 7560, 9240, 13860, 15120, 17550, 21840, 24024, 25200, 25740, 27720, 30030, 42504, 43680, 55440, 60060, 69300, 72072, 75600, 77520, 83160, 110880, 120120, 151200, 154440, 168168, 180180, 185640, 203490, 240240

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

			56 is a term, because it can be represented as a multinomial coefficient for 2 different partitions of 8 (and never for more than 2 different partitions of the same integer): 56 = 8!/(1!*1!*6!) = 8!/(3!*5!).

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

Second row of A376667.
Subsequence of A325306.
Cf. A036038, A078760, A376372, A376663.

A376670 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 3, i.e., numbers m such that A376663(m) = 3.

Original entry on oeis.org

166320, 360360, 720720, 2162160, 5045040, 5765760, 6683040, 7207200, 12252240, 14414400, 15135120, 24504480, 30270240, 35814240, 36756720, 37837800, 40360320, 40840800, 46558512, 49008960, 51482970, 61261200, 86486400, 98017920, 102965940, 110270160, 116396280

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

			166320 is a term, because it can be represented as a multinomial coefficient for 3 different partitions of 12 (and never for more than 3 different partitions of the same integer): 166320 = 12!/(1!*1!*1!*4!*5!) = 12!/(1!*1!*2!*2!*6!) = 12!/(2!*2!*3!*5!).

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

Third row of A376667.

Cf. A036038, A078760, A376373, A376663.

A376671 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 4, i.e., numbers m such that A376663(m) = 4.

Original entry on oeis.org

4084080, 17907120, 73513440, 75675600, 220540320, 411863760, 1102701600, 1210809600, 2162049120, 2205403200, 2327925600, 2471182560, 3087564480, 5145940800, 6983776800, 8380532160, 9777287520, 10291881600, 10296594000, 19554575040, 20583763200, 20593188000

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

			4084080 is a term, because it can be represented as a multinomial coefficient for 4 different partitions of 17 (and never for more than 4 different partitions of the same integer): 4084080 = 17!/(1!*1!*1!*4!*10!) = 17!/(1!*2!*5!*9!) = 17!/(2!*2!*3!*10!) = 17!/(4!*6!*7!).

Pontus von Brömssen, Table of n, a(n) for n = 1..9085 (all terms below 10^29)

Fourth row of A376667.

Cf. A036038, A078760, A376374, A376663.

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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A376664 Least number k such that there are A376663(n) partitions x_1 + ... + x_j = k such that the multinomial coefficient k!/(x_1! * ... * x_j!) is equal to n, i.e., the first row k of A036038 in which n appears A376663(n) times (or 0 if n = 0).

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A215911 G.f.: exp( Sum_{n>=1} A215910(n)*x^n/n ), where A215910(n) equals the sum of the n-th power of multinomial coefficients in row n of triangle A036038.

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A257468 Triangle read by rows in which the n-th row lists the multinomials A036038 for all partitions of 2n with only even parts in Abramowitz-Stegun ordering.

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A376668 Positive integers that do not appear more than once in the same row of A036038 (or A078760), i.e., numbers m such that A376663(m) = 1.

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A376669 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 2, i.e., numbers m such that A376663(m) = 2.

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A376670 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 3, i.e., numbers m such that A376663(m) = 3.

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A376671 Positive integers whose maximum frequency in a fixed row of A036038 (or A078760) is equal to 4, i.e., numbers m such that A376663(m) = 4.

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