A052295 -id:A052295 - cp's OEIS frontend

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

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

A060371 a(n) = (prime(n) - 1)! + 1.

Original entry on oeis.org

2, 3, 25, 721, 3628801, 479001601, 20922789888001, 6402373705728001, 1124000727777607680001, 304888344611713860501504000001, 265252859812191058636308480000001, 371993326789901217467999448150835200000001

Offset: 1

Jason Earls, Apr 01 2001

nonn

If the prime p is in A055469, that is if p = 2, 7, 11, 29, ... = A055469(j) which is valid for the first, 4th, 5th, 10th,.... entry here with j = 1, 2, 3, ..., then a(n) = A052295[A067186(j)] + 1. - R. J. Mathar, Apr 27 2007

It follows from Wilson's theorem that a(n) is divisible by the n-th prime. - Alonso del Arte, Feb 07 2014

Harry J. Smith, Table of n, a(n) for n = 1..88 (adapted by Vincenzo Librandi, Oct 17 2017)
Takashi Agoh, Karl Dilcher and Ladislav Skula, Wilson quotients for composite moduli, Math. Comp. 67 (1998), 843-861. MR 98h:11003.
C. K. Caldwell, Wilson Primes
R. Crandall, K. Dilcher and C. Pomerance, A search for Wieferich and Wilson primes, Math. Comp., 66 (1997), 433-449. MR 97c:11004.

Cf. A052295, A055469, A067186.

Subsequence of A038507. - Michel Marcus, Oct 17 2017

[Factorial(NthPrime(n)-1)+1: n in [1..15]]; // Vincenzo Librandi, Oct 17 2017

Table[(Prime[n] - 1)! + 1, {n, 12}] (* Alonso del Arte, Feb 07 2014 *)

{ n=1; forprime (p=1, 524, write("b060371.txt", n++, " ", (p - 1)! + 1); ) } \\ Harry J. Smith, Jul 04 2009

Corrected offset by Alonso del Arte, Feb 07 2014

A283261 Product of the different products of subsets of the set of numbers from 1 to n.

Original entry on oeis.org

1, 1, 2, 36, 331776, 42998169600000000, 13974055172471046820331520000000000000, 1833132881579690383668380351534446872452674453158326975200092938148249600000000000000000000000000

Offset: 0

Jaroslav Krizek, Mar 04 2017

nonn

Product of numbers in n-th row of A070861.

			Rows with subsets of the sets of numbers from 1 to n:
  {},
  {}, {1};
  {}, {1}, {2}, {1, 2};
  {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3};
  ...
Rows with the products of elements of these subsets:
  1;
  1, 1;
  1, 1, 2, 2;
  1, 1, 2, 3, 2, 3, 6, 6;
  ...
Rows with the different products of elements of these subsets:
  1;
  1;
  1, 2;
  1, 2, 3, 6;
  ...
a(0) = 1, a(1) = (1), a(2) = (1*2) = 2, a(3) = (1*2*3*6) = 36, ... .

Alois P. Heinz, Table of n, a(n) for n = 0..10

Cf. A000005, A000142, A027423, A060957, A070863, A052295, A263292.

b:= proc(n) option remember; `if`(n=0, {1},
      map(x-> [x, x*n][], b(n-1)))
    end:
a:= n-> mul(i, i=b(n)):
seq(a(n), n=0..7);  # Alois P. Heinz, Aug 01 2022

Table[Times @@ Union@ Map[Times @@ # &, Subsets@ Range@ n], {n, 7}] (* Michael De Vlieger, Mar 05 2017 *)

a(n)=my(v=[2..n]); factorback(Set(vector(2^(n-1),i, factorback(vecextract(v,i-1))))) \\ Charles R Greathouse IV, Mar 06 2017

a(n) <= n!^((A000005(n!))/2) = n!^(A027423(n)/2). - David A. Corneth, Mar 05 2017

a(n) = n!^(A263292(n)). - David A. Corneth, Mar 06 2017

a(0)=1 prepended by Alois P. Heinz, Aug 01 2022

A351884 Irregular triangle read by rows: T(n,k) is the number of sets of lists with distinct block sizes (as in A088311(n)) and containing exactly k lists.

Original entry on oeis.org

1, 0, 1, 0, 2, 0, 6, 6, 0, 24, 24, 0, 120, 240, 0, 720, 1440, 720, 0, 5040, 15120, 5040, 0, 40320, 120960, 80640, 0, 362880, 1451520, 1088640, 0, 3628800, 14515200, 14515200, 3628800, 0, 39916800, 199584000, 199584000, 39916800, 0, 479001600, 2395008000, 3353011200, 958003200

Offset: 0

Geoffrey Critzer, Feb 23 2022

			Triangle T(n,k) begins:
  1;
  0,     1;
  0,     2;
  0,     6,      6;
  0,    24,     24;
  0,   120,    240;
  0,   720,   1440,   720;
  0,  5040,  15120,  5040;
  0, 40320, 120960, 80640;
  ...

Columns k=0-1 give: A000007, A000142 (for n>=1).
Cf. A088311 (row sums).
T(A000217(n),n) gives A052295.
Cf. A003056, A293140.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+expand(x*b(n-i, min(i-1, n-i)))*n!/(n-i)!))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Feb 23 2022

nn = 13; Prepend[Map[Prepend[#, 0] &, Drop[Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[Series[Product[1 + y x^i, {i, 1, nn}], {x, 0, nn}],{x,y}]], 1]], {1}] // Grid

E.g.f.: Product_{i>=1} (1 + y*x^i).

Sum_{k=0..A003056(n)} (-1)^k * T(n,k) = A293140(n). - Alois P. Heinz, Feb 23 2022

A091478 Table of graphs with n (>=0) nodes and k (>=0) edges. Each type of object labeled from its own label set.

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 6, 6, 1, 6, 30, 120, 360, 720, 720, 1, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800, 3628800, 1, 15, 210, 2730, 32760, 360360, 3603600, 32432400, 259459200, 1816214400, 10897286400, 54486432000, 217945728000, 653837184000, 1307674368000, 1307674368000

Offset: 0

Christian G. Bower, Jan 13 2004

			  1;
  1;
  1, 1;
  1, 3,  6,   6;
  1, 6, 30, 120, 360, 720, 720;
  ...

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 114 (2.4.44).

Row sums: A091479.
Row lengths: A000124(n-1) for n>=1.
Columns 0-2: A000012, A000217(n-1), A033487(n-2).
a(n,A000217(n-1)) = A052295(n-1).
Cf. A006125, A008406.

a(n, k) = k!*binomial(binomial(n, 2), k).

T(0,0)=1 prepended by Alois P. Heinz, Feb 14 2023

A374293 a(n)/binomial(n,2)! is the probability that the minimum spanning tree of the complete graph of n vertices with i.i.d. random edge weights is a specific path.

Original entry on oeis.org

1, 1, 2, 44, 27120, 882241920, 2443792425984000, 846533597741050576896000, 50571850611494440562578575851520000, 686805008584962439650318114385825747697664000000, 2701735270674169239689693528384644314472371275610193920000000000, 3819958423456547324072333722421751679308286064300212197312630212725309440000000000

Offset: 1

Jamie Tucker-Foltz, Jul 02 2024

nonn

Equivalently, a(n) is the number of orderings of the edges of the complete graph of n vertices such that the minimal spanning tree (e.g., obtained by running Kruskal's algorithm with the edges in that order) is a specific path.

It appears that this is a subsequence of A253950. Specifically, a(n) appears at index m - n + 3, where m = binomial(n,2) is the number of edges of the complete graph on n vertices.

			a(3) = 2 because there are 2 orderings of the edges a, b, and c of K_3 that give the path {a, b}: (a, b, c) and (b, a, c).

Jamie Tucker-Foltz, Table of n, a(n) for n = 1..16
Eric Babson, Moon Duchin, Annina Iseli, Pietro Poggi-Corradini, Dylan Thurston, and Jamie Tucker-Foltz, Models of Random Spanning Trees, arXiv:2407.20226 [math.CO], 2024.
Jamie Tucker-Foltz, Code to compute a(n) on GitHub.

Cf. A052295, A253950.

E(p,m)={sum(k=0, m, sum(i=0, k, polcoef(p, i)*i!*(m-i)! )*x^k/(k!*(m-k)!))}
seq(n)={my(v=vector(n)); v[1]=1; for(n=2, n, my(p=sum(k=1, n-1, v[k]*v[n-k])); v[n]=E(intformal(p), binomial(n,2))); vector(n, n, my(m=binomial(n,2)); m!*polcoef(v[n], m))} \\ Andrew Howroyd, Jul 31 2024

cp's OEIS Frontend

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

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A060371 a(n) = (prime(n) - 1)! + 1.

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A283261 Product of the different products of subsets of the set of numbers from 1 to n.

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A351884 Irregular triangle read by rows: T(n,k) is the number of sets of lists with distinct block sizes (as in A088311(n)) and containing exactly k lists.

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A091478 Table of graphs with n (>=0) nodes and k (>=0) edges. Each type of object labeled from its own label set.

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Crossrefs

Formula

Extensions

A374293 a(n)/binomial(n,2)! is the probability that the minimum spanning tree of the complete graph of n vertices with i.i.d. random edge weights is a specific path.

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Author

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PARI

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