OFFSET
0,3
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
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
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) = 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
EXAMPLE
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)
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 *)
PROG
(PARI) a(n) = binomial(n+1, 2)!/prod(k=1, n, k^(n+1-k)); \\ Michel Marcus, May 02 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
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
STATUS
approved