This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A022915 #66 Feb 16 2025 08:32:34
%S A022915 1,1,3,60,12600,37837800,2053230379200,2431106898187968000,
%T A022915 73566121315513295589120000,65191584694745586153436251091200000,
%U A022915 1906765806522767212441719098019963758016000000,2048024348726152339387799085049745725891853852479488000000
%N A022915 Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!*1!*2!*...*n!).
%C A022915 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 A022915 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 A022915 Also the number of distinct adjacency matrices in the n-triangular honeycomb rook graph. - _Eric W. Weisstein_, Jul 14 2017
%H A022915 Alois P. Heinz, <a href="/A022915/b022915.txt">Table of n, a(n) for n = 0..35</a>
%H A022915 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AdjacencyMatrix.html">Adjacency Matrix</a>
%H A022915 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MultinomialCoefficient.html">Multinomial Coefficient</a>
%F A022915 a(n) = (n*(n+1)/2)!/(0!*1!*2!*...*n!).
%F A022915 a(n) = a(n-1) * A014068(n). - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Apr 08 2001.
%F A022915 a(n) = A052295(n)/A000178(n). - _Lekraj Beedassy_, Feb 19 2004
%F A022915 a(n) = A208437(n*(n+1)/2,n). - _Alois P. Heinz_, Apr 08 2016
%F A022915 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 A022915 a(n) = A327803(n*(n+1)/2,n). - _Alois P. Heinz_, Sep 25 2019
%F A022915 a(n) = A008480(A006939(n)). - _Gus Wiseman_, Aug 12 2020
%e A022915 From _Gus Wiseman_, Aug 12 2020: (Start)
%e A022915 The a(3) = 60 permutations of the prime indices of A006939(3) = 360:
%e A022915 (111223) (121123) (131122) (212113) (231211)
%e A022915 (111232) (121132) (131212) (212131) (232111)
%e A022915 (111322) (121213) (131221) (212311) (311122)
%e A022915 (112123) (121231) (132112) (213112) (311212)
%e A022915 (112132) (121312) (132121) (213121) (311221)
%e A022915 (112213) (121321) (132211) (213211) (312112)
%e A022915 (112231) (122113) (211123) (221113) (312121)
%e A022915 (112312) (122131) (211132) (221131) (312211)
%e A022915 (112321) (122311) (211213) (221311) (321112)
%e A022915 (113122) (123112) (211231) (223111) (321121)
%e A022915 (113212) (123121) (211312) (231112) (321211)
%e A022915 (113221) (123211) (211321) (231121) (322111)
%e A022915 (End)
%p A022915 with(combinat):
%p A022915 a:= n-> multinomial(binomial(n+1, 2), $0..n):
%p A022915 seq(a(n), n=0..12); # _Alois P. Heinz_, May 18 2013
%t A022915 Table[Apply[Multinomial ,Range[n]], {n, 0, 20}] (* _Geoffrey Critzer_, Dec 09 2012 *)
%t A022915 Table[Multinomial @@ Range[n], {n, 0, 20}] (* _Eric W. Weisstein_, Jul 14 2017 *)
%t A022915 Table[Binomial[n + 1, 2]!/BarnesG[n + 2], {n, 0, 20}] (* _Eric W. Weisstein_, Jul 14 2017 *)
%t A022915 Table[Length[Permutations[Join@@Table[i,{i,n},{i}]]],{n,0,4}] (* _Gus Wiseman_, Aug 12 2020 *)
%o A022915 (PARI) a(n) = binomial(n+1,2)!/prod(k=1, n, k^(n+1-k)); \\ _Michel Marcus_, May 02 2019
%Y A022915 Cf. A000178, A014068, A022919, A052295, A208437, A327803.
%Y A022915 A190945 counts the case of anti-run permutations.
%Y A022915 A317829 counts partitions of this multiset.
%Y A022915 A325617 is the version for factorials instead of superprimorials.
%Y A022915 A006939 lists superprimorials or Chernoff numbers.
%Y A022915 A008480 counts permutations of prime indices.
%Y A022915 A181818 gives products of superprimorials, with complement A336426.
%Y A022915 Cf. A000142, A022559, A027423, A034841, A076954, A112798, A303279, A336417.
%K A022915 nonn,easy
%O A022915 0,3
%A A022915 _Clark Kimberling_
%E A022915 More terms from Larry Reeves (larryr(AT)acm.org), Apr 11 2001
%E A022915 More terms from _Michel ten Voorde_, Apr 12 2001
%E A022915 Better definition from _L. Edson Jeffery_, May 18 2013