A027423 -id:A027423 - cp's OEIS frontend

A008302 Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_{i=0..n-1} (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). Also enumerates permutations by their major index.

1, 1, 1, 1, 2, 2, 1, 1, 3, 5, 6, 5, 3, 1, 1, 4, 9, 15, 20, 22, 20, 15, 9, 4, 1, 1, 5, 14, 29, 49, 71, 90, 101, 101, 90, 71, 49, 29, 14, 5, 1, 1, 6, 20, 49, 98, 169, 259, 359, 455, 531, 573, 573, 531, 455, 359, 259, 169, 98, 49, 20, 6, 1, 1, 7, 27, 76, 174, 343, 602, 961, 1415, 1940, 2493, 3017, 3450, 3736, 3836, 3736, 3450, 3017, 2493, 1940, 1415, 961, 602, 343, 174, 76, 27, 7, 1, 1, 8, 35, 111, 285, 628, 1230, 2191, 3606, 5545, 8031, 11021, 14395, 17957, 21450, 24584, 27073, 28675, 29228, 28675, 27073, 24584, 21450, 17957, 14395, 11021, 8031, 5545, 3606, 2191, 1230, 628, 285, 111, 35, 8, 1

Offset: 1

N. J. A. Sloane

T(n,k) is the number of permutations of {1..n} with k inversions.
n-th row gives growth series for symmetric group S_n with respect to transpositions (1,2), (2,3), ..., (n-1,n).
T(n,k) is the number of permutations of (1,2,...,n) having disorder equal to k. The disorder of a permutation p of (1,2,...,n) is defined in the following manner. We scan p from left to right as often as necessary until all its elements are removed in increasing order, scoring one point for each occasion on which an element is passed over and not removed. The disorder of p is the number of points scored by the end of the scanning and removal process. For example, the disorder of (3,5,2,1,4) is 8, since on the first scan, 3,5,2 and 4 are passed over, on the second, 3,5 and 4 and on the third scan, 5 is once again not removed. - Emeric Deutsch, Jun 09 2004
T(n,k) is the number of permutations p=(p(1),...,p(n)) of {1..n} such that Sum_{i: p(i)>p(i+1)} = k (k is called the Major index of p). Example: T(3,0)=1, T(3,1)=2, T(3,2)=2, T(3,3)=1 because the major indices of the permutations (1,2,3), (2,1,3), (3,1,2), (1,3,2), (2,3,1) and (3,2,1) are 0,1,1,2,2 and 3, respectively. - Emeric Deutsch, Aug 17 2004
T(n,k) is the number of 2 X c matrices with column totals 1,2,3,...,n and row totals k and binomial(n+1,2) - k. - Mitch Harris, Jan 13 2006
T(n,k) is the number of permutations p of {1,2,...,n} for which den(p)=k. Here den is the Denert statistic, defined in the following way: let p=p(1)p(2)...p(n) be a permutation of {1,2,...,n}; if p(i)>i, then we say that i is an excedance of p; let i_1 < i_2 < ... < i_k be the excedances of p and let j_1 < j_2 < ... < j_{n-k} be the non-excedances of p; let Exc(p) = p(i_1)p(i_2)...p(i_k), Nexc(p)=p(j_1)p(j_2)...p(j_{n-k}); then, by definition den(p) = i_1 + i_2 + ... + i_k + inv(Exc(p)) + inv(Nexc(p)), where inv denotes "number of inversions". Example: T(4,5)=3 because we have 1342, 3241 and 4321. We show that den(4321)=5: the excedances are 1 and 2; Exc(4321)=43, Nexc(4321)=21; now den(4321) = 1 + 2 + inv(43) + inv(21) = 3+1+1 = 5. - Emeric Deutsch, Oct 29 2008
T(n,k) is the number of size k submultisets of the multiset {1,2,2,3,3,3,...,n-1} (which contains i copies of i for 0 < i < n).
The limit of products of the numbers of fixed necklaces of length n composed of beads of types N(n,b), n --> infinity, is the generating function for inversions (we must exclude one unimportant factor b^n/n!). The error is < (b^n/n!)*O(1/n^(1/2-epsilon)). See Gaichenkov link. - Mikhail Gaichenkov, Aug 27 2012
The number of ways to distribute k-1 indistinguishable balls into n-1 boxes of capacity 1,2,3,...,n-1. - Andrew Woods, Sep 26 2012
Partial sums of rows give triangle A161169. - András Salamon, Feb 16 2013
The number of permutations of n that require k pair swaps in the bubble sort to sort them into the natural 1,2,...,n order. - R. J. Mathar, May 04 2013
Also series coefficients of q-factorial [n]q ! -- see Mathematica line. - _Wouter Meeussen, Jul 12 2014
From Mikhail Gaichenkov, Aug 16 2016: (Start)
Following asymptotic expansions in the Central Limit Theorem developed by Valentin V. Petrov, the cumulative distribution function of these numbers, CDF_N(x), is equal to the CDF of the normal distribution - (0.06/sqrt(2*Pi))*exp(-x^2/2)(x^3-3x)*(6N^3+21N^2+31N+31)/(N(2N+5)^2(N-1)+O(1/N^2).
This can be written as: CDF of the normal distribution -(0.09/(N*sqrt(2*Pi)))*exp(-x^2/2)*He_3(x) + O(1/N^2), N > 1, natural numbers (Gaichenkov, private research).
According to B. H. Margolius, Permutations with inversions, J. Integ. Seqs. Vol. 4 (2001), #01.2.4, "the unimodal behavior of the inversion numbers suggests that the number of inversions in a random permutation may be asymptotically normal". See links.
Moreover, E. Ben-Naim (Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory), "On the Mixing of Diffusing Particles" (13 Oct 2010), states that the Mahonian Distribution becomes a function of a single variable for large numbers of element, i.e., the probability distribution function is normal. See links.
To be more precise the expansion of the distribution is presented for a finite number of elements (or particles in terms of E. Ben-Naim's article). The distribution tends to the normal distribution for an infinite numbers of elements.
(End)
T(n,k) statistic counts (labeled) permutation graphs with n vertices and k edges. - Mikhail Gaichenkov, Aug 20 2019
From Gus Wiseman, Aug 12 2020: (Start)
Number of divisors of A006939(n - 1) or A076954(n - 1) with k prime factors, counted with multiplicity, where A006939(n) = Product_{i = 1..n} prime(i)^(n - i + 1). For example, row n = 4 counts the following divisors:
  1   2   4   8  24  72 360
      3   6  12  36 120
      5   9  18  40 180
         10  20  60
         15  30  90
             45
Crossrefs:
A336420 is the case with distinct prime multiplicities.
A006939 lists superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A317829 counts factorizations of superprimorials.
A336941 counts divisor chains under superprimorials.
Cf. A000005, A001222, A008278, A022559, A027423, A076954, A181796.
(End)
Named after the British mathematician Percy Alexander MacMahon (1854-1929). - Amiram Eldar, Jun 13 2021
Row maxima ~ n!/(sigma * sqrt(2*Pi)), sigma^2 = (2*n^3 + 9*n^2 + 7*n)/72 = variance of group type A_n (see also A161435). - Mikhail Gaichenkov, Feb 08 2023
Sum_{i>=0} T(n,i)*k^i = A069777(n,k). - Geoffrey Critzer, Feb 26 2025

			1; 1+x; (1+x)*(1+x+x^2) = 1+2*x+2*x^2+x^3; etc.
Triangle begins:
  n\k| 0  1   2    3    4     5     6     7     8      9     10
  ---+--------------------------------------------------------------
   1 | 1;
   2 | 1, 1;
   3 | 1, 2,  2,   1;
   4 | 1, 3,  5,   6,   5,    3,    1;
   5 | 1, 4,  9,  15,  20,   22,   20,   15,    9,     4,     1;
   6 | 1, 5, 14,  29,  49,   71,   90,  101,  101,    90,    71, ...
   7 | 1, 6, 20,  49,  98,  169,  259,  359,  455,   531,   573, ...
   8 | 1, 7, 27,  76, 174,  343,  602,  961, 1415,  1940,  2493, ...
   9 | 1, 8, 35, 111, 285,  628, 1230, 2191, 3606,  5545,  8031, ...
  10 | 1, 9, 44, 155, 440, 1068, 2298, 4489, 8095, 13640, 21670, ...
From _Gus Wiseman_, Aug 12 2020: (Start)
Row n = 4 counts the following submultisets of {1,1,1,2,2,3}:
  {}  {1}  {11}  {111}  {1112}  {11122}  {111223}
      {2}  {12}  {112}  {1122}  {11123}
      {3}  {22}  {122}  {1113}  {11223}
           {13}  {113}  {1123}
           {23}  {123}  {1223}
                 {223}
(End)

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Alois P. Heinz, Rows n = 1..50, flattened (first 30 rows from Paul D. Hanna)
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Thomas Wieder, Comments on A008302.
Wikipedia, Major index

Diagonals: A000707 (k=n-1), A001892 (k=n-2), A001893 (k=n-3), A001894 (k=n-4), A005283 (k=n-5), A005284 (k=n-6), A005285 (k=n-7).
Columns: A005286 (k=3), A005287 (k=4), A005288 (k=5), A242656 (k=6), A242657 (k=7).
Rows: A161435 (n=4), A161436 (n=5), A161437 (n=6), A161438 (n=7), A161439 (n=8), A161456 (n=9), A161457 (n=10).
Row-maxima: A000140, truncated table: A060701, row sums: A000142, row lengths: A000124.
A001809 gives total Denert index of all permutations.
A357611 gives a refinement.
Cf. A000217, A139365, A069777.

g := proc(n,k) option remember; if k=0 then return(1) else if (n=1 and k=1) then return(0) else if (k<0 or k>binomial(n,2)) then return(0) else g(n-1,k)+g(n,k-1)-g(n-1,k-n) end if end if end if end proc; # Barbara Haas Margolius (margolius(AT)math.csuohio.edu), May 31 2001
BB:=j->1+sum(t^i, i=1..j): for n from 1 to 8 do Z[n]:=sort(expand(simplify(product(BB(j), j=0..n-2)))) od: for n from 1 to 8 do seq(coeff(Z[n], t, j), j=0..(n-1)*(n-2)/2) od; # Zerinvary Lajos, Apr 13 2007
# alternative Maple program:
b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
       add(b(u+j-1, o-j)*x^(u+j-1), j=1..o)+
       add(b(u-j, o+j-1)*x^(u-j), j=1..u)))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=1..10);  # Alois P. Heinz, May 02 2017

f[n_] := CoefficientList[ Expand@ Product[ Sum[x^i, {i, 0, j}], {j, n}], x]; Flatten[Array[f, 8, 0]]
(* Second program: *)
T[0, 0] := 1; T[-1, k_] := 0;
T[n_, k_] := T[n, k] = If[0 <= k <= n*(n - 1)/2, T[n, k - 1] + T[n - 1, k] - T[n - 1, k - n], 0]; (* Peter Kagey, Mar 18 2021; corrected the program by Mats Granvik and Roger L. Bagula, Jun 19 2011 *)
alternatively (versions 7 and up):
Table[CoefficientList[Series[QFactorial[n,q],{q,0,n(n-1)/2}],q],{n,9}] (* Wouter Meeussen, Jul 12 2014 *)
b[u_, o_] := b[u, o] = Expand[If[u + o == 0, 1,
   Sum[b[u + j - 1, o - j]*x^(u + j - 1), {j, 1, o}] +
   Sum[b[u - j, o + j - 1]*x^(u - j), {j, 1, u}]]];
T[n_] := With[{p = b[n, 0]}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Apr 21 2025, after Alois P. Heinz *)

{T(n,k) = my(A=1+x); for(i=1,n, A = 1 + intformal(A - q*subst(A,x,q*x +x^2*O(x^n)))/(1-q)); polcoeff(n!*polcoeff(A,n,x),k,q)}
for(n=1,10, for(k=0,n*(n-1)/2, print1(T(n,k),", ")); print("")) \\ Paul D. Hanna, Dec 31 2016

row(n)=Vec(prod(k=1,n,(1-'q^k)/(1-'q))); \\ Joerg Arndt, Apr 13 2019

from sage.combinat.q_analogues import q_factorial
for n in (1..6): print(q_factorial(n).list()) # Peter Luschny, Jul 18 2016

Bourget, Comtet and Moritz-Williams give recurrences.
Mendes and Stanley give g.f.'s.
G.f.: Product_{j=1..n} (1-x^j)/(1-x) = Sum_{k=0..M} T{n, k} x^k, where M = n*(n-1)/2.
From Andrew Woods, Sep 26 2012, corrected by Peter Kagey, Mar 18 2021: (Start)
T(1, 0) = 1,
T(n, k) = 0 for n < 0, k < 0 or k > n*(n-1)/2.
T(n, k) = Sum_{j=0..n-1} T(n-1, k-j),
T(n, k) = T(n, k-1) + T(n-1, k) - T(n-1, k-n). (End)
E.g.f. satisfies: A(x,q) = 1 + Integral (A(x,q) - q*A(q*x,q))/(1-q) dx, where A(x,q) = Sum_{n>=0} x^n/n! * Sum_{k=0..n*(n-1)/2} T(n,k)*q^k, when T(0,0) = 1 is included. - Paul D. Hanna, Dec 31 2016

There were some mistaken edits to this entry (inclusion of an initial 1, etc.) which I undid. - N. J. A. Sloane, Nov 30 2009

Added mention of "major index" to definition. - N. J. A. Sloane, Feb 10 2019

A060775 The greatest divisor d|n such that d < n/d, with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 4, 3, 2, 1, 4, 1, 2, 3, 4, 1, 5, 1, 4, 3, 2, 5, 4, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 6, 1, 5, 3, 4, 1, 6, 5, 7, 3, 2, 1, 6, 1, 2, 7, 4, 5, 6, 1, 4, 3, 7, 1, 8, 1, 2, 5, 4, 7, 6, 1, 8, 3, 2, 1, 7, 5, 2, 3

Offset: 1

Labos Elemer, Apr 26 2001

Also: Largest divisor of n which is less than sqrt(n).
If n is not a square, then a(n) = A033676(n), else a(n) is strictly smaller than A033676(n) = sqrt(n) (except for a(1) = 1).  - M. F. Hasler, Sep 20 2011
Record values occur for n = k * (k+1), for which a(n) = k. - Franklin T. Adams-Watters, May 01 2015
If we define a divisor d|n to be strictly inferior if d < n/d, then strictly inferior divisors are counted by A056924 and listed by A341674. This sequence gives the greatest strictly inferior divisor, which may differ from the lower central divisor A033676. Central divisors are listed by A207375. - Gus Wiseman, Feb 28 2021

			n = 252, D = {1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252}, 18 divisors, the 9th is 14, so a(252) = 14.
From _Gus Wiseman_, Feb 28 2021: (Start)
The strictly inferior divisors of selected n:
n = 1  2  6  12  20  30  42  56  72  90  110  132  156  182  210  240
    -----------------------------------------------------------------
    {} 1  1  1   1   1   1   1   1   1   1    1    1    1    1    1
          2  2   2   2   2   2   2   2   2    2    2    2    2    2
             3   4   3   3   4   3   3   5    3    3    7    3    3
                     5   6   7   4   5   10   4    4    13   5    4
                                 6   6        6    6         6    5
                                 8   9        11   12        7    6
                                                             10   8
                                                             14   10
                                                                  12
                                                                  15
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms n = 2..1000 from Harry J. Smith)

Cf. A033677, A000196, A000005, A000142, A027423, A055226, A060776, A060777, A002378.
The weakly inferior version is A033676.
Positions of first appearances are A180291.
These are the row-maxima of A341674.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A070039 adds up strictly inferior divisors.
A207375 lists central divisors.
A333805 counts strictly inferior odd divisors.
A333806 counts strictly inferior prime divisors.
A341596 counts strictly inferior squarefree divisors.
A341677 counts strictly inferior prime-power divisors.
- Inferior: A063962, A066839, A069288, A161906, A217581, A333749, A333750.
- Superior: A051283, A059172, A063538, A063539, A070038, A161908, A341591.
- Strictly Superior: A048098, A064052, A140271, A238535, A341642, A341673.
Cf. A000203, A001248, A006530, A020639, A112798, A161901.

with(numtheory):
a:= n-> max(select(d-> is(d=1 or dAlois P. Heinz, Jan 29 2018

Table[Part[Divisors[w], Floor[DivisorSigma[0, w]/2]], {w, 1, 256}]
Table[If[n==1,1,Max[Select[Divisors[n],#Gus Wiseman, Feb 28 2021 *)

A060775(n)=if(n>1,divisors(n)[numdiv(n)\2],1) \\ M. F. Hasler, Sep 21 2011

a(n) = max { d: d|n and d < sqrt(n) or d = 1 }, where "|" means "divides". [Corrected by M. F. Hasler, Apr 03 2019]

a(1) = 1 added (to preserve the relation a(n) | n) by Franklin T. Adams-Watters, Jan 27 2018
Edited by M. F. Hasler, Apr 03 2019
Name changed by Gus Wiseman, Feb 28 2021 (was: Lower central (median) divisor of n, with a(1) = 1.)

A079210 Positive divisors of n!, listed in increasing order for each n, a new row for each n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 3, 6, 1, 2, 3, 4, 6, 8, 12, 24, 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720

Offset: 0

Christian van den Bosch (cjb(AT)cjb.ie), Jan 03 2003

Elements per row: 1,1,2,4,8,16,30,... (given by A027423, number of positive divisors of n!)

This sequence is the same as A070861 for the first 38 terms, but differs thereafter.

			First few rows are:
1;
1;
1,2;
1,2,3,6;
1,2,3,4,6,8,12,24;
1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120;
...

Andrew Howroyd, Table of n, a(n) for n = 0..1979 (rows 0..12)

Cf. A027423 (row lengths), A062569 (row sums), A070861.

[Divisors(Factorial(n)): n in [0..10]]; // Vincenzo Librandi, Jun 19 2015

Flatten[Table[Divisors[n!],{n,6}]]  (* Harvey P. Dale, Mar 13 2011 *)

tabf(nn) = for (n=0, nn, print(divisors(n!))); \\ Michel Marcus, Jun 19 2015

a(0)=1 prepended by Andrew Howroyd, Jan 26 2022

A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!1!2!...n!).

Original entry on oeis.org

1, 1, 3, 60, 12600, 37837800, 2053230379200, 2431106898187968000, 73566121315513295589120000, 65191584694745586153436251091200000, 1906765806522767212441719098019963758016000000, 2048024348726152339387799085049745725891853852479488000000

Offset: 0

Clark Kimberling

Number of ways to put numbers 1, 2, ..., n*(n+1)/2 in a triangular array of n rows in such a way that each row is increasing. Also number of ways to choose groups of 1, 2, 3, ..., n-1 and n objects out of n*(n+1)/2 objects. - Floor van Lamoen, Jul 16 2001
a(n) is the number of ways to linearly order the multiset {1,2,2,3,3,3,...n,n,...n}. - Geoffrey Critzer, Mar 08 2009
Also the number of distinct adjacency matrices in the n-triangular honeycomb rook graph. - Eric W. Weisstein, Jul 14 2017

			From _Gus Wiseman_, Aug 12 2020: (Start)
The a(3) = 60 permutations of the prime indices of A006939(3) = 360:
  (111223)  (121123)  (131122)  (212113)  (231211)
  (111232)  (121132)  (131212)  (212131)  (232111)
  (111322)  (121213)  (131221)  (212311)  (311122)
  (112123)  (121231)  (132112)  (213112)  (311212)
  (112132)  (121312)  (132121)  (213121)  (311221)
  (112213)  (121321)  (132211)  (213211)  (312112)
  (112231)  (122113)  (211123)  (221113)  (312121)
  (112312)  (122131)  (211132)  (221131)  (312211)
  (112321)  (122311)  (211213)  (221311)  (321112)
  (113122)  (123112)  (211231)  (223111)  (321121)
  (113212)  (123121)  (211312)  (231112)  (321211)
  (113221)  (123211)  (211321)  (231121)  (322111)
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..35
Eric Weisstein's World of Mathematics, Adjacency Matrix
Eric Weisstein's World of Mathematics, Multinomial Coefficient

Cf. A000178, A014068, A022919, A052295, A208437, A327803.
A190945 counts the case of anti-run permutations.
A317829 counts partitions of this multiset.
A325617 is the version for factorials instead of superprimorials.
A006939 lists superprimorials or Chernoff numbers.
A008480 counts permutations of prime indices.
A181818 gives products of superprimorials, with complement A336426.
Cf. A000142, A022559, A027423, A034841, A076954, A112798, A303279, A336417.

with(combinat):
a:= n-> multinomial(binomial(n+1, 2), $0..n):
seq(a(n), n=0..12);  # Alois P. Heinz, May 18 2013

Table[Apply[Multinomial ,Range[n]], {n, 0, 20}]  (* Geoffrey Critzer, Dec 09 2012 *)
Table[Multinomial @@ Range[n], {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
Table[Binomial[n + 1, 2]!/BarnesG[n + 2], {n, 0, 20}] (* Eric W. Weisstein, Jul 14 2017 *)
Table[Length[Permutations[Join@@Table[i,{i,n},{i}]]],{n,0,4}] (* Gus Wiseman, Aug 12 2020 *)

a(n) = binomial(n+1,2)!/prod(k=1, n, k^(n+1-k)); \\ Michel Marcus, May 02 2019

a(n) = (n*(n+1)/2)!/(0!*1!*2!*...*n!).
a(n) = a(n-1) * A014068(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001.
a(n) = A052295(n)/A000178(n). - Lekraj Beedassy, Feb 19 2004
a(n) = A208437(n*(n+1)/2,n). - Alois P. Heinz, Apr 08 2016
a(n) ~ A * exp(n^2/4 + n + 1/6) * n^(n^2/2 + 7/12) / (2^((n+1)^2/2) * Pi^(n/2)), where A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, May 02 2019
a(n) = A327803(n*(n+1)/2,n). - Alois P. Heinz, Sep 25 2019
a(n) = A008480(A006939(n)). - Gus Wiseman, Aug 12 2020

More terms from Larry Reeves (larryr(AT)acm.org), Apr 11 2001
More terms from Michel ten Voorde, Apr 12 2001
Better definition from L. Edson Jeffery, May 18 2013

A336416 Number of perfect-power divisors of n!.

Original entry on oeis.org

1, 1, 1, 1, 3, 3, 7, 7, 11, 18, 36, 36, 47, 47, 84, 122, 166, 166, 221, 221, 346, 416, 717, 717, 1001, 1360, 2513, 2942, 4652, 4652, 5675, 5675, 6507, 6980, 13892, 17212, 20408, 20408, 39869, 45329, 51018, 51018, 68758, 68758, 105573, 138617, 284718, 284718, 338126, 421126

Offset: 0

Gus Wiseman, Jul 22 2020

nonn

A number is a perfect power iff it is 1 or its prime exponents (signature) are not relatively prime.

			The a(1) = 0 through a(9) = 18 divisors:
       1: 1
       2: 1
       6: 1
      24: 1,4,8
     120: 1,4,8
     720: 1,4,8,9,16,36,144
    5040: 1,4,8,9,16,36,144
   40320: 1,4,8,9,16,32,36,64,128,144,576
  362880: 1,4,8,9,16,27,32,36,64,81,128,144,216,324,576,1296,1728,5184

David A. Corneth, Table of n, a(n) for n = 0..9999

The maximum among these divisors is A090630, with quotient A251753.
The version for distinct prime exponents is A336414.
The uniform version is A336415.
Replacing factorials with Chernoff numbers (A006939) gives A336417.
Prime powers are A000961.
Perfect powers are A001597, with complement A007916.
Prime power divisors are counted by A022559.
Cf. A000005, A001221, A001222, A011371, A032741, A118914, A124010, A130091, A327527.
Factorial numbers: A000142, A007489, A027423, A048656, A071626, A181796, A325272, A325273, A325617.

perpouQ[n_]:=Or[n==1,GCD@@FactorInteger[n][[All,2]]>1];
Table[Length[Select[Divisors[n!],perpouQ]],{n,0,15}]

a(n) = sumdiv(n!, d, (d==1) || ispower(d)); \\ Michel Marcus, Aug 19 2020

addhelp(val, "exponent of prime p in n!")
val(n, p) = my(r=0); while(n, r+=n\=p);r
a(n) = {if(n<=3, return(1)); my(pr = primes(primepi(n\2)), v = vector(#pr, i, val(n, pr[i])), res = 1, cv); for(i = 2, v[1], if(issquarefree(i), cv = v\i; res-=(prod(i = 1, #cv, cv[i]+1)-1)*(-1)^omega(i) ) ); res } \\ David A. Corneth, Aug 19 2020

a(p) = a(p-1) for prime p. - David A. Corneth, Aug 19 2020

a(26)-a(34) from Jinyuan Wang, Aug 19 2020

a(35)-a(49) from David A. Corneth, Aug 19 2020

A062569 a(n) = sigma(n!).

Original entry on oeis.org

1, 1, 3, 12, 60, 360, 2418, 19344, 159120, 1481040, 15334088, 184009056, 2217441408, 31044179712, 442487616480, 6686252969760, 107004539285280, 1926081707135040, 34683832925921088, 693676658518421760, 13891399238731734720, 292460416142501376000

Offset: 0

Jason Earls, Jul 03 2001

			a(4) = 60, since the sum of the positive divisors of 4! is 1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60. - _Timothy L. Tiffin_, Jan 22 2023

Wacław Sierpiński, Elementary Theory of Numbers, Ex. 6, p. 169, Warsaw, 1964.

Paul D. Hanna, Table of n, a(n) for n = 0..300
Rafael Jakimczuk, Two topics in number theory: sum of divisors of the factorial and a formula for primes, International Mathematical Forum, Vol. 12, No. 19 (2017), pp. 929-935. See Theorem 1.4, p. 932.
Vaclav Kotesovec, Plot of a(n)/(n!*log(n)) for n = 2..50000.

Cf. A000142, A000203, A027423, A073004.

with(numtheory):seq(sigma(n!), n=0..19); # Zerinvary Lajos, Feb 15 2008

Array[DivisorSigma[1,#! ]&,33,1] (* Vladimir Joseph Stephan Orlovsky, Nov 01 2009 *)

PARI
```
for(n=0,21,print(sigma(n!)))
```

[sigma(ZZ(n).factorial(), 1) for n in range(20)]  # Zerinvary Lajos, Jun 13 2009

a(n) = A000203(A000142(n)). - Michel Marcus, Jan 10 2015
a(p) = (p+1)*a(p-1) for p prime. - Altug Alkan, Jul 18 2016
Limit_{n->oo} a(n)/n! = oo. Proof in Sierpiński. - Bernard Schott, Feb 09 2019
a(n) ~ c * n! * log(n) * (1 + O(1/log(n))), where c =  exp(gamma) = A073004 (Jakimczuk, 2017). - Amiram Eldar, Nov 07 2020

A336414 Number of divisors of n! with distinct prime multiplicities.

Original entry on oeis.org

1, 1, 2, 3, 7, 10, 20, 27, 48, 86, 147, 195, 311, 390, 595, 1031, 1459, 1791, 2637, 3134, 4747, 7312, 10766, 12633, 16785, 26377, 36142, 48931, 71144, 82591, 112308, 128023, 155523, 231049, 304326, 459203, 568095, 642446, 812245, 1137063, 1441067, 1612998, 2193307, 2429362

Offset: 0

Gus Wiseman, Jul 22 2020

nonn

A number has distinct prime multiplicities iff its prime signature is strict.

			The first and second columns below are the a(6) = 20 counted divisors of 6! together with their prime signatures. The third column shows the A000005(6!) - a(6) = 10 remaining divisors.
      1: ()      20: (2,1)    |    6: (1,1)
      2: (1)     24: (3,1)    |   10: (1,1)
      3: (1)     40: (3,1)    |   15: (1,1)
      4: (2)     45: (2,1)    |   30: (1,1,1)
      5: (1)     48: (4,1)    |   36: (2,2)
      8: (3)     72: (3,2)    |   60: (2,1,1)
      9: (2)     80: (4,1)    |   90: (1,2,1)
     12: (2,1)  144: (4,2)    |  120: (3,1,1)
     16: (4)    360: (3,2,1)  |  180: (2,2,1)
     18: (1,2)  720: (4,2,1)  |  240: (4,1,1)

Chai Wah Wu, Table of n, a(n) for n = 0..6245 (n = 0..94 from David A. Corneth)

Perfect-powers are A001597, with complement A007916.
Numbers with distinct prime multiplicities are A130091.
Divisors with distinct prime multiplicities are counted by A181796.
The maximum divisor with distinct prime multiplicities is A327498.
Divisors of n! with equal prime multiplicities are counted by A336415.
Cf. A000005, A098859, A118914, A124010, A327527, A336424, A336500, A336568.
Factorial numbers: A000142, A007489, A022559, A027423, A048656, A048742, A071626, A325272, A325273, A325617, A336416.

Table[Length[Select[Divisors[n!],UnsameQ@@Last/@FactorInteger[#]&]],{n,0,15}]

a(n) = sumdiv(n!, d, my(ex=factor(d)[,2]); #vecsort(ex,,8) == #ex); \\ Michel Marcus, Jul 24 2020

a(n) = A181796(n!).

a(21)-a(41) from Alois P. Heinz, Jul 24 2020

A008278 Reflected triangle of Stirling numbers of 2nd kind, S(n,n-k+1), n >= 1, 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 7, 1, 1, 10, 25, 15, 1, 1, 15, 65, 90, 31, 1, 1, 21, 140, 350, 301, 63, 1, 1, 28, 266, 1050, 1701, 966, 127, 1, 1, 36, 462, 2646, 6951, 7770, 3025, 255, 1, 1, 45, 750, 5880, 22827, 42525, 34105, 9330, 511, 1

Offset: 1

N. J. A. Sloane

The n-th row also gives the coefficients of the sigma polynomial of the empty graph \bar K_n. - Eric W. Weisstein, Apr 07 2017
The n-th row also gives the coefficients of the independence polynomial of the (n-1)-triangular honeycomb bishop graph. - Eric W. Weisstein, Apr 03 2018
From Gus Wiseman, Aug 11 2020: (Start)
Conjecture: also the number of divisors of the superprimorial A006939(n - 1) that have 0 <= k <= n distinct prime factors, all appearing with distinct multiplicities. For example, row n = 4 counts the following divisors of 360:
  1  2  12  360
     3  18
     4  20
     5  24
     8  40
     9  45
        72
Equivalently, T(n,k) is the number of length-n vectors 0 <= v_i <= i with k nonzero values, all of which are distinct.
Crossrefs:
A006939 lists superprimorials or Chernoff numbers.
A022915 counts permutations of prime indices of superprimorials.
A076954 can be used instead of A006939.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A336420 is the version counting all prime factors, not just distinct ones.
Cf. A000005, A027423, A095149, A124010, A317829, A327498, A336419, A336421.
(End)
From Leonidas Liponis, Aug 26 2024: (Start)
It appears that this sequence is related to the combinatorial form of Faà di Bruno's formula. Specifically, the number of terms for the n-th derivative of a composite function y = f(g(x)) matches the number of partitions of n.
For example, consider the case where g(x) = e^x, in which all derivatives of g(x) are equal. The first 5 rows of A008278 appear as the factors of derivatives of f(x), highlighted here in brackets:
          dy/dx = [ 1 ] * f'(e^x) * e^x
          d^2y/dx^2 = [ 1 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x
          d^3y/dx^3 = [ 1 ] * f'''(e^x) * e^{3x} + [ 3 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x
          d^4y/dx^4 = [ 1 ] * f''''(e^x) * e^{4x} + [ 6 ] * f'''(e^x) * e^{3x} + [ 7 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x
          d^5y/dx^5 = [ 1 ] * f'''''(e^x) * e^{5x} + [ 10 ] * f''''(e^x) * e^{4x} + [ 25 ] * f'''(e^x) * e^{3x} + [ 15 ] * f''(e^x) * e^{2x} + [ 1 ] * f'(e^x) * e^x
This pattern is observed in Mathematica for the first 10 cases, using the code below.
(End)

			The e.g.f. of [0,0,1,7,25,65,...], the k=3 column of A008278, but with offset n=0, is exp(x)*(1*(x^2)/2! + 4*(x^3)/3! + 3*(x^4)/4!).
Triangle starts:
  1;
  1,  1;
  1,  3,   1;
  1,  6,   7,    1;
  1, 10,  25,   15,    1;
  1, 15,  65,   90,   31,    1;
  1, 21, 140,  350,  301,   63,    1;
  1, 28, 266, 1050, 1701,  966,  127,   1;
  1, 36, 462, 2646, 6951, 7770, 3025, 255, 1;
  ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 835.
F. N. David, M. G. Kendall and D. E. Barton, Symmetric Function and Allied Tables, Cambridge, 1966, p. 223.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994.

T. D. Noe, Rows n = 0..100 of triangle, flattened
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].
Noureddine Chair, Exact two-point resistance, and the simple random walk on the complete graph minus N edges, Ann. Phys. 327, No. 12, 3116-3129 (2012), eq. (27).
Xi Chen, Bishal Deb, Alexander Dyachenko, Tomack Gilmore, and Alan D. Sokal, Coefficientwise total positivity of some matrices defined by linear recurrences, arXiv:2012.03629 [math.CO], 2020.
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015
U. N. Katugampola, A new Fractional Derivative and its Mellin Transform, arXiv preprint arXiv:1106.0965 [math.CA], 2011.
Eric Weisstein's World of Mathematics, Bell Polynomial
Eric Weisstein's World of Mathematics, Empty Graph
Eric Weisstein's World of Mathematics, Independence Polynomial
Eric Weisstein's World of Mathematics, Sigma Polynomial
Eric Weisstein's World of Mathematics, Stirling Number of the Second Kind

See A008277 and A048993, which are the main entries for this triangle of numbers.

Cf. A094262, A008277, A008276, A003422, A000166, A000110, A000204, A000045, A000108, A213735.

a008278 n k = a008278_tabl !! (n-1) !! (k-1)
a008278_row n = a008278_tabl !! (n-1)
a008278_tabl = iterate st2 [1] where
  st2 row = zipWith (+) ([0] ++ row') (row ++ [0])
            where row' = reverse $ zipWith (*) [1..] $ reverse row
-- Reinhard Zumkeller, Jun 22 2013

rows = 10; Flatten[Table[StirlingS2[n, k], {n, 1, rows}, {k, n, 1, -1}]] (* Jean-François Alcover, Nov 17 2011, *)
Table[CoefficientList[x^n BellB[n, 1/x], x], {n, 10}] // Flatten (* Eric W. Weisstein, Apr 05 2017 *)
n = 5; Grid[Prepend[Transpose[{Range[1, n], Table[D[f[Exp[x]], {x, i}], {i, 1, n}]}], {"Order","Derivative"}], Frame -> All, Spacings -> {2, 1}] (* Leonidas Liponis, Aug 27 2024 *)

for(n=1,10,for(k=1,n,print1(stirling(n,n-k+1,2),", "))) \\ Hugo Pfoertner, Aug 30 2020

T(n, k)=0 if n < k, T(n, 0)=0, T(1, 1)=1, T(n, k) = (n-k+1)*T(n-1, k-1) + T(n-1, k) otherwise.
O.g.f. for the k-th column: 1/(1-x) if k=1 and A(k,x):=((x^k)/(1-x)^(2*k+1))*Sum_{m=0..k-1} A008517(k,m+1)*x^m if k >= 2. A008517 is the second-order Eulerian triangle. Cf. p. 257, eq. (6.43) of the R. L. Graham et al. book. - Wolfdieter Lang, Oct 14 2005
E.g.f. for the k-th column (with offset n=0): E(k,x):=exp(x)*Sum_{m=0..k-1} A112493(k-1,m)*(x^(k-1+m))/(k-1+m)! if k >= 1. - Wolfdieter Lang, Oct 14 2005
a(n) = abs(A213735(n-1)). - Hugo Pfoertner, Sep 07 2020

Name edited by Gus Wiseman, Aug 11 2020

cp's OEIS Frontend

A379928 Numbers m such that A379742(m) is a multiple of A027423(m).

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A008302 Triangle of Mahonian numbers T(n,k): coefficients in expansion of Product_{i=0..n-1} (1 + x + ... + x^i), where k ranges from 0 to A000217(n-1). Also enumerates permutations by their major index.

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A060775 The greatest divisor d|n such that d < n/d, with a(1) = 1.

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A079210 Positive divisors of n!, listed in increasing order for each n, a new row for each n.

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A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!*1!*2!*...*n!).

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A336416 Number of perfect-power divisors of n!.

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A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!1!2!...n!).