%I #65 Jul 14 2023 10:14:43

%S 1,1,3,60,12600,37837800,2053230379200,2431106898187968000,

%T 73566121315513295589120000,65191584694745586153436251091200000,

%U 1906765806522767212441719098019963758016000000,2048024348726152339387799085049745725891853852479488000000

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

%C 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

%C 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

%C Also the number of distinct adjacency matrices in the n-triangular honeycomb rook graph. - _Eric W. Weisstein_, Jul 14 2017

%H Alois P. Heinz, <a href="/A022915/b022915.txt">Table of n, a(n) for n = 0..35</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AdjacencyMatrix.html">Adjacency Matrix</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MultinomialCoefficient.html">Multinomial Coefficient</a>

%F a(n) = (n*(n+1)/2)!/(0!*1!*2!*...*n!).

%F a(n) = a(n-1) * A014068(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001.

%F a(n) = A052295(n)/A000178(n). - _Lekraj Beedassy_, Feb 19 2004

%F a(n) = A208437(n*(n+1)/2,n). - _Alois P. Heinz_, Apr 08 2016

%F 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

%F a(n) = A327803(n*(n+1)/2,n). - _Alois P. Heinz_, Sep 25 2019

%F a(n) = A008480(A006939(n)). - _Gus Wiseman_, Aug 12 2020

%e From _Gus Wiseman_, Aug 12 2020: (Start)

%e The a(3) = 60 permutations of the prime indices of A006939(3) = 360:

%e (111223) (121123) (131122) (212113) (231211)

%e (111232) (121132) (131212) (212131) (232111)

%e (111322) (121213) (131221) (212311) (311122)

%e (112123) (121231) (132112) (213112) (311212)

%e (112132) (121312) (132121) (213121) (311221)

%e (112213) (121321) (132211) (213211) (312112)

%e (112231) (122113) (211123) (221113) (312121)

%e (112312) (122131) (211132) (221131) (312211)

%e (112321) (122311) (211213) (221311) (321112)

%e (113122) (123112) (211231) (223111) (321121)

%e (113212) (123121) (211312) (231112) (321211)

%e (113221) (123211) (211321) (231121) (322111)

%e (End)

%p with(combinat):

%p a:= n-> multinomial(binomial(n+1, 2), $0..n):

%p seq(a(n), n=0..12); # _Alois P. Heinz_, May 18 2013

%t Table[Apply[Multinomial ,Range[n]], {n, 0, 20}] (* _Geoffrey Critzer_, Dec 09 2012 *)

%t Table[Multinomial @@ Range[n], {n, 0, 20}] (* _Eric W. Weisstein_, Jul 14 2017 *)

%t Table[Binomial[n + 1, 2]!/BarnesG[n + 2], {n, 0, 20}] (* _Eric W. Weisstein_, Jul 14 2017 *)

%t Table[Length[Permutations[Join@@Table[i,{i,n},{i}]]],{n,0,4}] (* _Gus Wiseman_, Aug 12 2020 *)

%o (PARI) a(n) = binomial(n+1,2)!/prod(k=1, n, k^(n+1-k)); \\ _Michel Marcus_, May 02 2019

%Y Cf. A000178, A014068, A022919, A052295, A208437, A327803.

%Y A190945 counts the case of anti-run permutations.

%Y A317829 counts partitions of this multiset.

%Y A325617 is the version for factorials instead of superprimorials.

%Y A006939 lists superprimorials or Chernoff numbers.

%Y A008480 counts permutations of prime indices.

%Y A181818 gives products of superprimorials, with complement A336426.

%Y Cf. A000142, A022559, A027423, A034841, A076954, A112798, A303279, A336417.

%K nonn,easy

%O 0,3

%A _Clark Kimberling_

%E More terms from Larry Reeves (larryr(AT)acm.org), Apr 11 2001

%E More terms from _Michel ten Voorde_, Apr 12 2001

%E Better definition from _L. Edson Jeffery_, May 18 2013