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A022915
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Multinomial coefficients (0, 1, ..., n)! = C(n+1,2)!/(0!*1!*2!*...*n!).
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37
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1, 1, 3, 60, 12600, 37837800, 2053230379200, 2431106898187968000, 73566121315513295589120000, 65191584694745586153436251091200000, 1906765806522767212441719098019963758016000000, 2048024348726152339387799085049745725891853852479488000000
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OFFSET
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0,3
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COMMENTS
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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
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LINKS
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FORMULA
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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) ~ 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
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EXAMPLE
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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)
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MAPLE
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with(combinat):
a:= n-> multinomial(binomial(n+1, 2), $0..n):
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MATHEMATICA
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Table[Apply[Multinomial , Range[n]], {n, 0, 20}] (* Geoffrey Critzer, Dec 09 2012 *)
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 *)
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PROG
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(PARI) a(n) = binomial(n+1, 2)!/prod(k=1, n, k^(n+1-k)); \\ Michel Marcus, May 02 2019
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CROSSREFS
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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.
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Apr 11 2001
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STATUS
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approved
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