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

Author

Clark Kimberling

Keywords

• nonn
• easy

Comments

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

Examples

			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)
		

Links

• 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

Crossrefs

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.

Programs

Maple

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

Mathematica

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 *)

PARI

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

Formula

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

Extensions

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