A036038 -id:A036038 - cp's OEIS frontend

A048996 Irregular triangle read by rows. Preferred multisets: numbers refining A007318 using format described in A036038.

1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 3, 1, 1, 2, 2, 3, 3, 4, 1, 1, 2, 2, 1, 3, 6, 1, 4, 6, 5, 1, 1, 2, 2, 2, 3, 6, 3, 3, 4, 12, 4, 5, 10, 6, 1, 1, 2, 2, 2, 1, 3, 6, 6, 3, 3, 4, 12, 6, 12, 1, 5, 20, 10, 6, 15, 7, 1, 1, 2, 2, 2, 2, 3, 6, 6, 3, 3, 6, 1, 4, 12, 12, 12, 12, 4, 5, 20, 10, 30, 5, 6, 30, 20, 7, 21, 8, 1

Offset: 0

Alford Arnold

This array gives in row n>=1 the multinomial numbers (call them M_0 numbers) m!/product((a_j)!,j=1..n) with the exponents of the partitions of n with number of parts m:=sum(a_j,j=1..n), given in the Abramowitz-Stegun order. See p. 831 of the given reference. See also the arrays for the M_1, M_2 and M_3 multinomial numbers A036038, A036039 and A036040 (or A080575).
For a signed version see A111786.
These M_0 multinomial numbers give the number of compositions of n >= 1 with parts corresponding to the partitions of n (in A-St order). See an n = 5 example below. The triangle with the summed entries of like number of parts m is A007318(n-1, m-1) (Pascal). - Wolfdieter Lang, Jan 29 2021

			Table starts:
[1]
[1]
[1, 1]
[1, 2, 1]
[1, 2, 1, 3, 1]
[1, 2, 2, 3, 3, 4, 1]
[1, 2, 2, 1, 3, 6, 1, 4, 6,  5, 1]
[1, 2, 2, 2, 3, 6, 3, 3, 4, 12, 4, 5, 10, 6, 1]
.
T(5,6) = 4 because there are four multisets using the first four digits {0,1,2,3}: 32100, 32110, 32210 and 33210
T(5,6) = 4 because there are 4 compositions of 5 that can be formed from the partition 2+1+1+1. - _Geoffrey Critzer_, May 19 2013
These 4 compositions 2+1+1+1, 1+2+1+1, 1+1+2+1 and 1+1+1+2 of 5 correspond to the 4 set partitions of [5] :={1,2,3,4,5}, with 4 blocks of consecutive numbers, namely {1,2},{3},{4},{5} and {1},{2,3},{4},{5} and {1},{2},{3,4},{5} and {1},{2},{3},{4,5}. - _Wolfdieter Lang_, May 30 2018

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1972.
Wolfdieter Lang, First 10 rows and more.

Cf. A000670, A007318, A036035, A036038, A019538, A115621, A309004, A000079 (row sums), A000041 (row lengths).

nmax:=9: with(combinat): for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while sA036040(n, m) := (add(q(t), t=1..n))!/(mul(q(t)!, t=1..n)); od: od: seq(seq(A036040(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016

C(sig)={my(S=Set(sig)); (#sig)!/prod(k=1, #S, (#select(t->t==S[k], sig))!)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 18 2020

from collections import Counter
def ASPartitions(n, k):
    Q = [p.to_list() for p in Partitions(n, length=k)]
    for q in Q: q.reverse()
    return sorted(Q)
def A048996_row(n):
    h = lambda p: product(map(factorial, Counter(p).values()))
    return [factorial(len(p))//h(p) for k in (0..n) for p in ASPartitions(n, k)]
for n in (1..10): print(A048996_row(n)) # Peter Luschny, Nov 02 2019 [corrected on notice from Sean A. Irvine, Apr 30 2022]

T(n,k) = A036040(n,k) * Factorial(A036043(n,k)) / A036038(n,k) = A049019(n,k) / A036038(n,k).
If the n-th partition is P, a(n) is the multinomial coefficient of the signature of P. - Franklin T. Adams-Watters, May 30 2006
T(n,k) = A309004(A036035(n,k)). - Andrew Howroyd, Oct 19 2020

More terms from Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 17 2001

a(0)=1 prepended by Andrew Howroyd, Oct 19 2020

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.

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.

A183610 Rectangular table where T(n,k) is the sum of the n-th powers of the k-th row of multinomial coefficients in triangle A036038 for n>=0, k>=0, as read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 5, 10, 5, 1, 1, 9, 46, 47, 7, 1, 1, 17, 244, 773, 246, 11, 1, 1, 33, 1378, 15833, 19426, 1602, 15, 1, 1, 65, 8020, 354065, 1980126, 708062, 11481, 22, 1, 1, 129, 47386, 8220257, 221300626, 428447592, 34740805, 95503, 30

Offset: 0

Paul D. Hanna, Aug 11 2012

			The table begins:
n=0: [1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 77, 101, 135, 176, ...];
n=1: [1, 1, 3, 10, 47, 246, 1602, 11481, 95503, 871030, 8879558, ...];
n=2: [1, 1, 5, 46, 773, 19426, 708062, 34740805, 2230260741, ...];
n=3: [1, 1, 9, 244, 15833, 1980126, 428447592, 146966837193, ...];
n=4: [1, 1, 17, 1378, 354065, 221300626, 286871431922, ...];
n=5: [1, 1, 33, 8020, 8220257, 25688403126, 199758931567152, ...];
n=6: [1, 1, 65, 47386, 194139713, 3033434015626, 141528428949437282, ...];
n=7: [1, 1, 129, 282124, 4622599553, 361140600078126, ...];
n=8: [1, 1, 257, 1686178, 110507041025, 43166813000390626, ...];
n=9: [1, 1, 513, 10097380, 2646977660417, 5169878244001953126, ...];
n=10:[1, 1, 1025, 60525226, 63465359844353, 619778904740009765626, ...];
...
The sums of the n-th power of terms in row k of triangle A036038 begin:
T(n,1) = 1^n,
T(n,2) = 1^n + 2^n,
T(n,3) = 1^n + 3^n + 6^n,
T(n,4) = 1^n + 4^n + 6^n + 12^n + 24^n,
T(n,5) = 1^n + 5^n + 10^n + 20^n + 30^n + 60^n + 120^n,
T(n,6) = 1^n + 6^n + 15^n + 20^n + 30^n + 60^n + 90^n + 120^n + 180^n + 360^n + 720^n, ...
Note that row n=0 forms the partition numbers A000041.

Paul D. Hanna, Table of rows 0..32 as read by antidiagonals, with index n = 0..528.

Cf. A036038, A000041, A005651, A183240, A183235, A183236, A183237, A183238, A215910 (main diagonal).

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      b(n-i, min(n-i, i), k)/i!^k+b(n, i-1, k))
    end:
A:= (n, k)-> k!^n*b(k$2, n):
seq(seq(A(d-k, k), k=0..d), d=0..10);  # Alois P. Heinz, Sep 11 2019

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, 1, b[n-i, Min[n-i, i], k]/i!^k + b[n, i-1, k]];
A[n_, k_] := k!^n b[k, k, n];
Table[Table[A[d-k, k], {k, 0, d}], {d, 0, 10}] // Flatten (* Jean-François Alcover, Dec 06 2019, after Alois P. Heinz *)

{T(n,k)=k!^n*polcoeff(1/prod(m=1, k, 1-x^m/m!^n +x*O(x^k)), k)}
for(n=0,10,for(k=0,8,print1(T(n,k),", "));print(""))

G.f. of row n: Sum_{k>=0} T(n,k)*x^k/k!^n = Product_{j>=1} 1/(1 - x^j/j!^n).

A215910 a(n) = sum of the n-th power of the multinomial coefficients in row n of triangle A036038.

Original entry on oeis.org

1, 1, 5, 244, 354065, 25688403126, 141528428949437282, 83257152559805973052807833, 7012360438832401192319979008881500417, 109324223115831487504443410090345278639832867784010, 396327911646787133737309113762487915762995734538047874429637296650

Offset: 0

Paul D. Hanna, Aug 26 2012

nonn

			The sums of the n-th power of multinomial coefficients in row n of triangle A036038 begin:
a(1) = 1^1 = 1;
a(2) = 1^2 + 2^2 = 5;
a(3) = 1^3 + 3^3 + 6^3 = 244;
a(4) = 1^4 + 4^4 + 6^4 + 12^4 + 24^4 = 354065;
a(5) = 1^5 + 5^5 + 10^5 + 20^5 + 30^5 + 60^5 + 120^5 = 25688403126;
a(6) = 1^6 + 6^6 + 15^6 + 20^6 + 30^6 + 60^6 + 90^6 + 120^6 + 180^6 + 360^6 + 720^6 = 141528428949437282;
a(7) = 1^7 + 7^7 + 21^7 + 35^7 + 42^7 + 105^7 + 140^7 + 210^7 + 210^7 + 420^7 + 630^7 + 840^7 + 1260^7 + 2520^7 + 5040^7 = 83257152559805973052807833; ...
which also form a logarithmic generating function of an integer series:
L(x) = x + 5*x^2/2 + 244*x^3/3 + 354065*x^4/4 + 25688403126*x^5/5 +...
where
exp(L(x)) = 1 + x + 3*x^2 + 84*x^3 + 88602*x^4 + 5137769389*x^5 +...+ A215911(n)*x^n +...

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

Cf. A183610, A036038, A036740, A005651, A183240, A183235, A183236, A183237, A183238; A215911.

Cf. A326321.

b:= proc(n, i, k) option remember; `if`(n=0 or i=1, 1,
      b(n-i, min(n-i, i), k)/i!^k+b(n, i-1, k))
    end:
a:= n-> n!^n*b(n$3):
seq(a(n), n=0..12);  # Alois P. Heinz, Sep 11 2019

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || i == 1, 1, b[n - i, Min[n - i, i], k]/i!^k + b[n, i - 1, k]];
a[n_] := n!^n b[n, n, n];
a /@ Range[0, 12] (* Jean-François Alcover, Nov 01 2020, after Alois P. Heinz *)

{a(n)=n!^n*polcoeff(1/prod(m=1, n, 1-x^m/m!^n +x*O(x^n)), n)}
for(n=1,15,print1(a(n),", "))

a(n) = [x^n/n!^n] * Product_{k=1..n} 1/(1 - x^k/k!^n) for n>=1, with a(0)=1.
Logarithmic derivative of A215911, ignoring the initial term a(0).
a(n) ~ (n!)^n = A036740(n). - Vaclav Kotesovec, Feb 19 2015
a(n) ~ 2^(n/2) * Pi^(n/2) * n^(n*(2*n+1)/2) / exp(n^2 - 1/12). - Vaclav Kotesovec, Feb 19 2015

A376673 Least number whose maximum frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., least number m such that A376663(m) = n, or 0 if no such number exists.

Original entry on oeis.org

1, 56, 166320, 4084080, 1396755360, 698377680, 146659312800, 1075501627200, 37104806138400, 3710480613840000, 296838449107200, 86825246363856000, 96472495959840000, 36466603472819520000, 35251050023725536000, 272194921062320256000, 408292381593480384000

Offset: 1

Pontus von Brömssen, Oct 02 2024

nonn

After a(36), the sequence continues (where "?" represents terms that are either 0 or greater than 10^29): ?, 3059734941813910128088320000, ?, ?, 64254433778092112689854720000. After a(41), all terms are either 0 or greater than 10^29.

The terms a(1), a(3), ..., a(15), a(24), a(26), ..., a(36), a(38), a(41) are all in A025487, but a(16), ..., a(23), a(25) are all divisible by 17^2 but not by 13^2.

			First few terms and their representations as multinomial coefficients (corresponding to partitions with sum A376664(n)):
  a(1) =          1 = 0!;
  a(2) =         56 = 8!/(1!*1!*6!) = 8!/(3!*5!);
  a(3) =     166320 = 12!/(1!*1!*1!*4!*5!) = 12!/(1!*1!*2!*2!*6!) = 12!/(2!*2!*3!*5!);
  a(4) =    4084080 = 17!/(1!*1!*1!*4!*10!) = 17!/(1!*2!*5!*9!) = 17!/(2!*2!*3!*10!) = 17!/(4!*6!*7!);
  a(5) = 1396755360 = 19!/(1!*1!*1!*1!*1!*4!*10!) = 19!/(1!*1!*1!*2!*5!*9!) = 19!/(1!*1!*2!*2!*3!*10!) = 19!/(1!*1!*4!*6!*7!) = 19!/(3!*4!*5!*7!).

Pontus von Brömssen, Table of n, a(n) for n = 1..36

First column of A376667.

Cf. A025487, A036038, A078760, A376376, A376663, A376664.

A138533 Resort the multinomial sequence A036038 by source partition as described in A126442, A129306 and A136101.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 6, 120, 60, 20, 5, 1, 30, 10, 720, 360, 120, 30, 6, 1, 180, 60, 15, 20, 90

Offset: 1

Alford Arnold, Mar 27 2008

Multinomials count permutations of multisets and also paths in lattices; for example, there are six paths (from null to full) through the lattice of divisors for signature 36: 2233 2323 2332 3223 3232 and 3322.

			a(11) is six because the eleventh least prime signature in source format is 36 the signature for partition 2+2 the ninth partition and A036038(9) = 6.
The tables begin:
1.......2.......6.......24......120.....720....5040.....40320......362880
........1.......3.......12.......60.....360....2520.....20160......181440
................1.......4........20.....120.....840......6720.......60480
........................1........5.......30.....210......1680.......15120
.. ..............................1........6......42......336........3024
..........................................1.......7.......56.........504
..................................................1........8..........72
...........................................................1...........9
.......................................................................1
........................6........30.....180....1260....10080........90720
.................................10......60.....420.....3360........30240
...

Cf. A000142 A001710 A005651 (column sums) A025487 A053445.

Cf. A173333. [From Reinhard Zumkeller, Feb 19 2010]

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.

cp's OEIS Frontend

A048996 Irregular triangle read by rows. Preferred multisets: numbers refining A007318 using format described in A036038.

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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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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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A183610 Rectangular table where T(n,k) is the sum of the n-th powers of the k-th row of multinomial coefficients in triangle A036038 for n>=0, k>=0, as read by antidiagonals.

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A215910 a(n) = sum of the n-th power of the multinomial coefficients in row n of triangle A036038.

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A376673 Least number whose maximum frequency in a fixed row of A036038 (or A078760) is equal to n, i.e., least number m such that A376663(m) = n, or 0 if no such number exists.

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A138533 Resort the multinomial sequence A036038 by source partition as described in A126442, A129306 and A136101.

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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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