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%I #22 Oct 16 2015 03:01:08
%S 0,0,3,21,72,180,375,693,1176,1872,2835,4125,5808,7956,10647,13965,
%T 18000,22848,28611,35397,43320,52500,63063,75141,88872,104400,121875,
%U 141453,163296,187572,214455,244125,276768,312576,351747,394485,441000
%N Ordered (2,2)-selections from the multiset {1,1,2,2,3,3,...,n,n}.
%C Number of ordered (2,2)-selections which can be taken from the first 2n elements of A008619, the positive integers repeated. Order does count among subselections, e.g. [[1,1],[2,2]] and [[2,2],[1,1]] are different (2,2)-selections. Order does not count within a subselection, e.g. [1,3] is equivalent to [3,1].
%C Many thanks to Alois P. Heinz, Joerg Arndt, and Olivier Gérard for pointing out bugs in earlier versions of this sequence and for their comments!
%C The number of (not ordered) (2,2)-selections from natural numbers repeated = A008619 is equal to A086602 (observed by Alois P. Heinz).
%C The number of ordered (1,1)-selections from natural numbers repeated = A008619 is equal to the squares = A000290.
%C The number of ordered (1,1)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the Oblong numbers = A002378.
%C The number of ordered (2,2)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to A033487.
%C The number of (not ordered) (1,1)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the triangular numbers = A000217).
%C The number of (not ordered) (2,2)-selections from the natural numbers = A000027 ("[1,2,3,...,n]-multiset") is equal to the tritriangular numbers = A050534.
%C For n>0, the terms of this sequence are related to A014209 by a(n) = sum( i*A014209(i), i=0..n-1 ). [_Bruno Berselli_, Dec 20 2013]
%H Vincenzo Librandi, <a href="/A188667/b188667.txt">Table of n, a(n) for n = 0..1000</a>
%H Quang T. Bach, Roshil Paudyal, Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1510.04310">A Fibonacci analogue of Stirling numbers</a>, arXiv:1510.04310 [math.CO], 2015 (see p. 28).
%H T. Wieder, <a href="http://dx.doi.org/10.3968/j.pam.1925252820110201.010">Generation of all possible multiselections from a multiset</a>, Progress in Applied Mathematics, 2(1) (2011), 61-66, DOI:10.3968/j.pam.1925252820110201.010. - Thomas Wieder, Oct 15 2011
%F a(n) = n*(n+4)*(n-1)^2/4.
%F G.f.: 3*x^2*(x^2-2*x-1) / (x-1)^5.
%e Example: For n=3 there are 21 ordered selections of the type (2,2):
%e [[1,1],[2,2]], [[1,2],[1,2]], [[2,2],[1,1]], [[1,2],[2,3]],
%e [[1,3],[2,2]], [[2,2],[1,3]], [[2,3],[1,2]], [[1,1],[2,3]],
%e [[1,2],[1,3]], [[1,3],[1,2]], [[2,3],[1,1]], [[1,1],[3,3]],
%e [[1,3],[1,3]], [[3,3],[1,1]], [[1,2],[3,3]], [[1,3],[2,3]],
%e [[2,3],[1,3]], [[3,3],[1,2]], [[2,2],[3,3]], [[2,3],[2,3]],
%e [[3,3],[2,2]].
%t Table[n*(n + 4)*(n - 1)^2/4, {n, 0, 100}] (* _Vincenzo Librandi_, Oct 18 2012 *)
%Y Cf. A014209.
%K nonn,easy
%O 0,3
%A _Thomas Wieder_, Apr 07 2011
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